Semiregular Large Sets By Charles A. Cusack A THESIS ... - CiteSeerX
Semiregular Large Sets
By Charles A. Cusack
A THESIS
Presented to the Faculty of The Graduate College at the University of Nebraska In Partial Fulfillment of Requirements For the Degree of Master of Science Major: Computer Science
Under the Supervision of Professor Spyros Magliveras
Lincoln, Nebraska
May, 1998
Acknowledgments I would like to start by thanking the faculty of the Department of Computer Science and Engineering and the Department of Mathematics and Statistics at UNL. I have learned many things in the last several years that have helped me in several areas of this thesis. Along with the faculty, I would like to thank the office staff who do much in the department that is unseen, and have certainly done much for me personally. I would also like to thank Phil Romig for teaching me too many things to mention. Also, I would like to thank the members of my committee, Professors Spyros Magliveras, Douglas Stinson, and Jean-Camille Birget. They have taken time out of their busy schedules to read and comment on drafts of the thesis. I would like to thank my adviser, Professor Spyros Magliveras. He has taught me many things in the last 3 years, and has sparked my interest in several areas of research. He encourages me both personally and professionally. Lastly, I would like to thank my Lord and Savior, Jesus Christ, who has been the constant in my life as a student, and will continue to be when I am no longer a student. He has revealed to me a another part of His Big Book of Theorems. He gives meaning and purpose to my life and work, and has saved me from that which I could not save myself. To God be the glory!
Semiregular Large Sets Charles A. Cusack, M.S. University of Nebraska, 1998
Adviser: Spyros Magliveras
A t − (v, k, λ) design (X, B) is a v-element set X of points and a collection B of kelement subsets of X called blocks such that every t-element subset of X is contained in precisely λ blocks. The collection design with λ=λ=
v−t k−t
X k
of all k-subsets of a set X forms a t-(v,k,λ)
, and is called the complete design.
An LS[N](t, k, v), or a large sets of t − (v, k, λ) designs, is a partition, [(X, Bi )]N i=1 , of the complete design into N disjoint t − (v, k, λ/N) designs. These are also called large sets of t − (v, k, λ) designs. A group G is said to be an automorphism group of a large set B if Bg =B for all g ∈ G. That is, if Big ∈ B for all Bi ∈ B and g ∈ G. Equivalently, we say that a large set with this property is G-invariant. If the stronger condition that Big =Bi for all Bi ∈ B and g ∈ G holds, we say that a large set is [G]-invariant. If the G-orbits of a set X are all of length |G|, we say that G acts semiregularly on X. In particular, if t < k, G is t-homogeneous on X, and G acts semiregularly
on the k-subsets of X, then there exists an LS[N](t, k, v). In this case, we say it is a G-semiregular large set. In chapter 2, we discuss transitive groups and give a few details about the group PSL(2, q). Then in chapter 3 we discuss the existence of PSL(2, q)-semiregular LS[N] (3,k,q+1). In chapter 4 we discuss the use of recursive constructions to find new large sets. In particular, we give the parameters for some large sets with small v that can be obtained using these constructions.
Contents
1 Introduction
1
1.1
Set and Number Theory . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.2
Design Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.3
Group Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.4
Large Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
2 Transitive Groups
14
2.1
Examples of transitive groups . . . . . . . . . . . . . . . . . . . . . .
14
2.2
A list of all t- and t∗ -transitive groups . . . . . . . . . . . . . . . . . .
16
2.3
PSL(2,q) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
3 The Existence of G-semiregular Large Sets
20
3.1
PSL(2,q) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
3.2
t- and t∗ -transitive groups with t ≥ 3 . . . . . . . . . . . . . . . . . .
27
i
4 Recursive Constructions for Large Sets
30
4.1
Known Recursive Constructions . . . . . . . . . . . . . . . . . . . . .
30
4.2
Applying the Recursive Constructions . . . . . . . . . . . . . . . . . .
32
5 Further Research
37
A Density of Large Sets from PSL(2,q)
39
ii
List of Figures 1.1
A graphical representation of a 2 − (7, 3, 1) design . . . . . . . . . . .
iii
5
List of Tables 2.1
Useful facts about PSL(2,q) when q ≡ 3 mod 4 . . . . . . . . . . . . .
18
2.2
Finite t-transitive and t∗ -transitive groups . . . . . . . . . . . . . . .
19
3.1
The density of p for which there exists a PSL(2,p)-semiregular large sets. 26
4.1
Large sets with v ≤ 30 . . . . . . . . . . . . . . . . . . . . . . . . . .
32
A.1 The density of p for which there exists a PSL(2,p)-semiregular large sets. 40
iv
Chapter 1 Introduction The theory of t − (v, k, λ) designs, or simply t−designs, is an important part of combinatorics. The question of existence of non-trivial t−designs for t > 5 was open until the early 1980’s. Motivated by the classification of finite simple groups, and the corollary that t−transitive groups do not exist for t > 5, some researchers thought that for t > 5, t-designs could not exist. This conjecture was proved wrong by Leavitt and Magliveras in 1983 [32, 33] when they constructed the first simple 6-designs from the exceptional 4-homogeneous group P ΓL2 (32) acting on the projective line of 33 points. Infinite families of 5-designs were not known to exist until 1972 [1]. In fact, with the exception of two 5-designs found by E. Kramer, all known 5-designs found before 1982 were on an even number of points. The combinatorial design research landscape changed dramatically in 1985 when L. Teirlinck [40] showed that simple t−designs exist for arbitrarily large t and v. It became obvious from Teirlinck’s 1
2 construction that the “high-road” to the construction of t−designs for large t is by means of the notion of Large Sets of t−designs. A large set constitutes a partition of the collection of all k−subsets of a given point set X into disjoint t − (v, k, λ) designs. In the last dozen years, several recursive constructions resulting in infinite families of large sets of t−designs have been discovered, and we think there are certainly more to come. These constructions have as corollaries the existence of t−designs with both small and large t, previously unknown to exist. To date, no large sets of t−designs are known to exist with t > 5 and λ = 1 (Steiner systems.) In fact, not even a single non-trivial t−design is known for t > 5 with λ = 1, let alone a large set of such designs. Further, even though infinitely many large sets of 3-designs are known, again, not a single large set of 3-(v,4,1) designs is known.
1.1
Set and Number Theory
Let X be a set. If X consists of n points, we say X has cardinality n, and denote this by |X|=n. If K ⊆ X has cardinality k, we sat that K is a k-subset of X. We denote the set of all k-subsets of a set X by
X k
. It is not hard to see that
X k
=
|X| k
.
Let x, y, and z be integers. We write x ≡ y mod z when x=y+wz for some integer w. The following theorem is well known.
3 Theorem 1.1 Chinese Remainder Theorem
Let m1 , m2 , . . . , mr denote r
positive integers that are pairwise relatively prime, let m = m1 m2 · · · mr , and let a1 , a2 , . . . , ar denote any r integers with 0 ≤ ai ≤ mi − 1. Then the system of r congruences x ≡ ai mod mi (1 ≤ i ≤ r) has a unique solution modulo m.
The following is a simple extension of the Chinese Remainder Theorem.
Corollary 1.1 Let r1 , r2 , . . . , rn be pairwise relatively prime positive integers, with r = r1 r2 · · · rn , and let t1 , t2 , . . . , tn be positive integers such that 0 ≤ ti ≤ ri for 1 ≤ i ≤ n. For each i = 1, . . . , n, let si1 , si2 , . . . , siti be distinct integers modulo ri . Then the set of relations
x ∈ {s11 , s12 , . . . , s1t1 } mod r1 , and x ∈ {s2 , s2 , . . . , s2 } mod r2 , and t2 1 2 .. . x ∈ {s , s , . . . , s } mod r n1 n2 nt n n has exactly t1 t2 · · · t3 distinct solutions modulo r.
The next two theorems relate to the distribution of primes.
Theorem 1.2 (Dirichlet) If gcd(a, k) = 1, then there are infinitely many primes p ≡ a mod k.
4 Theorem 1.3 The primes are uniformly distributed among the φ(k) residue classes modulo k that correspond to invertible elements in Zk .
For more information about number theory, see [34]. For an in-depth treatment of the distribution of primes, see chapter 7 of [2].
1.2
Design Theory
A simple t − (v, k, λ) design, (X, B), is a v-element set X of points and a collection B of k-element subsets of X called blocks, such that every t-element subset of X is contained in precisely λ blocks. All designs in this thesis will be simple. It is well known that for each 0 ≤ s ≤ t every t − (v, k, λ) design is also an s − (v, k, λs ) design, where k−s v−s . λs = λ t−s t−s Thus, a set of necessary conditions (called the divisibility conditions) for the existence of a t − (v, k, λ) design is that
v−s λ t−s
k−s t−s
be an integer for s = 0, 1, 2, . . . , t − 1. We denote by λ∗ (t, k, v) (or simply λ∗ , if the parameter set is clear) the minimum positive integer λ that satisfies the divisibility
5 conditions. The number of blocks in the design is k v . b = λ0 = λ t t Example: A 2 − (7, 3, 1) design is given by X = {0, 1, 2, 3, 4, 5, 6}, and B={013, 124, 235, 346, 450, 561, 602}. Figure 1.1 shows a graphical representation of the design. The blocks are the 6 lines and 1 circle. This design is the smallest projective plane, known as the Fano plane. 0
1
5 2
3
6
4
Figure 1.1: A graphical representation of a 2 − (7, 3, 1) design
For a thorough treatment of design theory, see [16].
1.3
Group Theory
For basic definitions in group theory, see [35]. We denote by SX the symmetric group on the set X. Let X be a set, let G be a group, and suppose the formal multiplication
X ×G→X
(x, g) 7→ xg
6 is defined and satisfies the following axioms:
1. x1 = x for all x ∈ X, where 1 is the identity in G. 2. (x)gh = (xg )h for all x ∈ X, and for all g, h ∈ G.
Then we say that G acts on the set X, and we write G|X. If H ⊆ G, we define xH ={xh : h ∈ H}. The G-orbit of x, or simply the orbit of x, is the set xG . The stabilizer of x ∈ X in G is Gx = {g ∈ G : xg = x}. It is not hard to see that Gx is a subgroup of G. A group action G|X is called semiregular if all G-orbits of X are of length |G|. A group action G|X is called transitive if X consists of one G-orbit (i.e., xG =X for all x ∈ X). Let G be a group and let x ∈ G. The conjugate of x by g ∈ G is the element g −1 xg. Let X=G. Then G acts on X by conjugation, i.e. xg = g −1 xg. The orbit xG of x ∈ G in this action is called the conjugacy class of x. The stabilizer Gx = {g ∈ G : g −1 xg = x} is the set of elements that commute with x, and is called the centralizer of x in G. It is usually denoted by CG (x). If G|X is a group action and g ∈ G, we define Fix(g) by Fix(g)={x ∈ X|xg = x}. We denote |Fix(g)| by χG|X (g), or simply χ(g) if the group action is understood.
7 It is elementary to establish the following results (see [15, 16, 17, 18, 35]): Theorem 1.4 |xG | = [G : Gx ] = |G|/|Gx|. Theorem 1.5 The number of conjugates of x in G is [G : CG (g)] = |G|/|CG (x)|. Theorem 1.6 Gxg =(Gx )g . Theorem 1.7 Fix(g h)=(Fix(g))h, and therefore χ(g h )=χ(g). The following elementary result is due to Cauchy and Frobenius. It is commonly but erroneously referred to as Burnside’s lemma. Theorem 1.8 (Cauchy-Frobenius) Let a group G act on a finite set X. If N is the number of orbits of G, then N=
1 X χ(g). |G| g∈G
Let k be a positive integer. Then the action G|X induces an action G| obvious way: If x = {x1 , x2 , . . . , xk } ∈
X k
X k
in the
and g ∈ G, then xg = {xg1 , xg2 , . . . , xgk }.
Similarly, G acts on the k-tuples of X in a natural way: If ~x = (x1 , x2 , . . . , xk ) is a k-tuple of X, then ~xg = (xg1 , xg2 , . . . , xgk ). The group action G|X is said to be t-homogeneous if the action of G on the tsubsets of X is transitive. That is, G|X is t-homogeneous if for any pair of t-element subsets S1 , S2 of X, there is an element g ∈ G such that S1g = S2 .
8 The group G is said to be t-transitive on X if the induced action of G on the t-tuples of X is transitive. That is, if for any pair of t-tuples ~x, ~y of X, there is an element g ∈ G such that ~xg = ~y . A group is said to be of type t∗ if it is t-homogeneous, but not t-transitive. For simplicity, we call these t∗ -transitive. The subgroup G of SX is an automorphism group of a design (X, B) if for all g ∈ G and S ∈ B, S g ∈ B. In this case, we say the t-design is G-invariant. If S ⊆ X, the orbit of S is S G = {S g : g ∈ G} and the stabilizer of S is GS = {g ∈ G : S g = S}. It follows that G is an automorphism group of a design (X, B) if and only if B is a union of G-orbits on
X k
.
A system of distinct orbit representatives of the orbits that make up B is called a collection of base blocks. Example: It is evident from Figure 1 that the group G = h(034)(165), (15)(34)i is an automorphism group of the 2 − (7, 3, 1) design given in Section 1.2. Furthermore, under the group G, {013, 124, 156} are base blocks. Let t < k, and let G act on X. Let ∆ = {∆i }ri=1 be the G-orbits on Γ = {Γj }rj=1 be the G-orbits on
X k
X t
, and let
. For each ∆i , fix an arbitrary orbit representative
Ti . Let A∆,Γ = (ai,j ) be the matrix in which entry ai,j is the number of elements of
9 Γj which contain Ti . Although perhaps not immediately obvious, it is true that any representative from ∆i will be contained in the same number of elements of Γj . The following theorem is due to Kramer and Mesner.
Theorem 1.9 There exists a G-invariant t-(v,k,λ) if and only if there is a {0, 1}solution x to the matrix equation
A∆,Γ · x = λ(1, 1, . . . , 1)⊤ .
In other words, to find a t-design, one must find a set of columns of A∆,Γ whose row-sums are all equal to λ. Notice that if G is t-homogeneous, then there is only one orbit on t-subsets of X. Thus the matrix A∆,Γ has only one row. It follows that if G is a t-homogeneous group, then for any k ≥ t, any orbit of k-element subsets of G forms a t-(v,k,λ) design. Moreover, any union of such orbits is also a t-design. We denote by GF(q) the finite field of order q, where q is a prime power. The non-zero entries of a finite field form a multiplicative cyclic group GF(q)∗ of order q − 1. A primitive element (or primitive root) α ∈ GF(q) is an element that generates GF(q)∗ . For more information on group theory, the reader is directed to [35]. For more on automorphism groups of a design, see [5, 7].
10
1.4
Large Sets
The collection
X k
of all k-subsets of a set X forms a t-(v,k,λ) design with λ=λ=
v−t k−t
.
This is usually called the complete design. An LS[N](t, k, v), or a large set of t − (v, k, λ) designs, is a partition, [(X, Bi )]N i=1 , of the complete design into N disjoint t − (v, k, λ/N) designs. Large sets are also denoted LSλ (t, k, v) to place the emphasis on the value of λ. The λ is omitted if it is one. Recall that λ∗ is the minimal value of λ for which there exists a t−(v, k, λ) design. A large set of t − (v, k, λ∗ ) designs is called a (t, k, v)-decomposition. It should be noted that the original definition of a large set was restricted to only those for which λ = λ∗ . Let B = {Bi }N i=1 be the collection of designs in a large set. A group G is said to be an automorphism group of a large set if Bg =B for all g ∈ G, that is, if Big ∈ B for all Bi ∈ B and g ∈ G. Equivalently, we say that a large set with this property is G-invariant. If the stronger condition that Big =Bi for all Bi ∈ B and g ∈ G holds, we say that a large set is [G]-invariant. Let G be t-homogeneous on X with a semiregular induced action on
X k
. Then
the matrix A∆,Γ has only one row, and all entries are the same. Since each orbit is a
11 t-(v,k,λ) design, it is clear that there is an LS[N](t,k,v) with N=
v k
/|G|. In this case
we say the large set is G-semiregular. It is not too hard to see that a G-semiregular large set is [G]-invariant, and that a [G]-invariant large set is G-invariant. Theorems 1.10, 1.11, 1.12, and 1.13 give necessary and sufficient conditions for the existence of all three classes of large sets. Theorem 1.10 Let G be t-homogeneous on X. Then there exists a G-semiregular large set if and only if no element of prime order fixes any k-subsets of X. Proof. Let G be t-homogeneous on X. Assume that no element of prime order fixes any k-subsets of X. Then, if B ∈
X k
then GB =1 otherwise, if 1 6= g ∈ GB , there is
a power of g which is an element of prime order fixing B. Hence, |B G | =|G|, and the collection of all G-orbits of k-subsets is a G-semiregular large set. Conversely, if B={Bi }ri=1 is a semiregular large set of t − (v, k, λ) designs, then each Bi is a G−orbit of size |G|, and therefore if B ∈ Bi , we have GB = 1, so no elements of prime order fix B.
The following two theorems allude to two different techniques which can be used to find [G]-invariant large sets. The first is a simple extension of Theorem 1.9. Essentially it says that a [G]-invariant large set exists if and only if one can find N disjoint solutions to the matrix equations given in Theorem 1.9.
12 Theorem 1.11 A [G]-invariant LS[N]-(t,k,v) exists if and only if there is a {0, 1}solution Y with constant row sum 1 to the 1 .. A∆,Γ · Y = λ . 1
matrix equation 1 ... 1 .. .. . . 1 ... 1
|∆|,N
Theorem 1.12 essentially states that if all t-designs with a G as an automorphism group are known, then one needs simply to find a collection of these that partitions the orbits on k-subsets to find a [G]-invariant large set. The vector x selects the disjoint t-designs. Theorem 1.12 Let Y be a matrix whose rows are all solutions to the matrix equation given in Theorem 1.9. Then a [G]-invariant LS[N]-(t,k,v) exists if and only if there is a {0, 1}-solution x to the matrix equation xY = (1, 1, . . . , 1) Denote the orbits of
X k
under the action of G by Γ = {Γi : 1 ≤ i ≤ s}. Also, let
Θ = {Θi}ri=1 be the G-orbits on the collection of all t − (v, k, λ) designs with point set X. For each Θi , fix an arbitrary orbit representative Di . Define the r × s matrix M = mij by the rule mij = |Di ∩ Γj | × |Θi|/|Γj |. It should be noted that the entry mij is independent of the orbit representative Di that is chosen. Theorem 1.13 gives the necessary and sufficient conditions for the existence of G-invariant large sets.
13 Theorem 1.13 There exists a G-invariant large set of t − (v, k, λ) designs if and only if there exists a {0, 1}-solution x to the matrix equation xM = (1, 1, . . . , 1)⊤ .
Tables of the existence of t-designs and large sets for v ≤ 30 are given in [9], and in [27], the existence of large sets is completely determined for v ≤ 12. A survey on large sets is given by Teirlinck in [16]. In [11], the authors list all known parameter sets for large sets with v ≤ 18. The table indicates whether a large set with the given parameters exists, doesn’t exist, or is unknown.
Chapter 2 Transitive Groups 2.1
Examples of transitive groups
Let q = pa be a prime power, let X = GF (q), let V = V (n, q) be a vector space of dimension n over GF(q), and let N be an arbitrary set. Also, In is defined to be the n × n identity matrix. When the size is understood, it will be denoted simply by I. In the remainder of this section, we define some of the groups used in the remainder of the thesis. The definitions of the following groups, and more information about them, can be found in many sources, including Rotman’s text [35].
• SN is the symmetric group on the set N.
• AN , called the alternating group, is the subgroup of SN consisting of all even permutations.
14
15 • GL(n,q), the general linear group, is the multiplicative group of all nonsingular m × m matrices over GF (q).
• ZGL (n, q), the center of GL(n,q), is the subgroup consisting of all non-zero scalar transformations.
• SL(n,q), the special linear group, is the multiplicative group of all n×n matrices over GF (q) with determinant 1.
• ZSL (n, q), the center of SL(n,q), is the group of all n × n matrices of the form kI, with k n = 1.
• ΓL(n, q) is the group of all nonsingular semilinear transformations of V to itself under composition. The center of ΓL(n, q) is ZGL (n, q). • PGL(n,q) = GL(n, q)/ZGL(n, q), the projective linear group. • PSL(n,q) = SL(n, q)/ZSL (n, q), the projective special linear group, also called the projective unimodular group.
• PΓL(n,q) = ΓL(n, q)/ZGL (n, q), the collineation group. The following definitions can be found in [21].
16 • AΓL(1,q) is the group of mappings x 7→ axσ + b on GF(q), where a 6= 0 and b are in GF(q) and σ ∈ Aut(GF(q)).
• AGL(1,q), the affine group, is the group of mappings x 7→ ax + b on GF(q), where a 6= 0, and b are in GF(q).
Carter [8] gives descriptions of the following groups, including many references.
• PSU(3,q) is the projective special unitary group.
• PSp(2d, 2) is the projective symplectic group. This is a subgroup of PSL(2d, q).
• 2 B2 (q) is the Suzuki group. • 2 G2 (q) is the Ree group.
• HS is the Higman-Sims group.
• Co3 is the third Conway group.
• M11 , M12 , M22 , M23 , and M24 are the Mathieu groups.
2.2
A list of all t- and t∗-transitive groups
It is clear that any t-transitive group is t-homogeneous. The fact that for t ≥ 5 the converse is true was proved by Livingstone and Wagner [31]. Thus for t ≥ 5, there are
17 no t∗ -transitive groups. Also, if G is a t-transitive group of degree n, then G is also (n − t)-homogeneous. Thus, when speaking of t∗ -transitive groups, we will assume that n ≥ 2t. The following theorem due to Kantor [21] determines all t∗ -transitive groups.
Theorem 2.1 Let G be a group of type t∗ on a finite set of n points, where n ≥ 2t. Then up to permutation isomorphism, one of the following holds:
(i) t=2 and G ≤ AΓL(1, q) with n = q ≡ 3 mod 4;
(ii) t=3 and PSL(2, q) ≤ G ≤ P ΓL(2, q), where n − 1 = q ≡ 3 mod 4;
(iii) t=3 and G = AGL(1, 8), AΓL(1, 8) or AΓL(1, 32); or
(iv) t=4 and G = P SL(2, 8), P ΓL(2, 8) or P ΓL(2, 32).
All t- and t∗ -transitive groups are essentially known. Lists of 2-transitive groups are widely available [4, 22, 6]. Cameron [6] gives a list that includes enough information to classify all t- and t∗ -transitive groups. More precisely, all t- and t∗ -transitive groups are known for t > 3. 3- or 3∗ -transitive groups must have PSL(2,q) as a minimal normal subgroup, with a few exceptions. For t = 2, there is a longer list of minimal normal subgroups. Table 2.2 gives the characterization of t and t∗ -transitive
18 groups. Note that a group listed as t-transitive is not (necessarily) also listed as (t − 1)-transitive.
2.3
PSL(2,q)
The action of PSL(2,q)on the projective line X = GF(q) ∪ {∞} is well known (when q ≡ 3 mod 4 see [13], for example). The results below will be of use.
Theorem 2.2 PSL(2,q)is 3-homogeneous on X when q ≡ 3 mod 4.
Theorem 2.3 Let g ∈ PSL(2,q)be an element of order d. If g has an a-cycle, then a = d or a = 1.
Table 2.1 contains information about the elements of PSL(2,q). In the table, φ(n) is the number of numbers mod n relatively prime to n. 1
p
2
d| q−1 2
d| q+1 , d 6= 2 2
order of the centralizer of g
(q 3 −q) 2
q
q+1
(q−1) 2
(q+1) 2
number of conjugacy classes
1
2
1
φ(d)/2
φ(d)/2
q2 − 1
(q 2 −q)
φ(d)(q 2 +q)
2
2
φ(d)(q 2 −q) 2
1
0
2
0
order of g
number of elements
1
number of fixed points χ(g) q + 1
Table 2.1: Useful facts about PSL(2,q) when q ≡ 3 mod 4
19
If G is t- or t∗ -transitive on a set X, then: t Group degree t ≥ 6 G = Sk is k transitive k G = Ak is (k − 2)-transitive k A6 has two representations t∗ , t ≥ 5 There are none 5 G = M24 24 G = M12 (Two representations) 12 4 G = M23 23 G = M11 11 ∗ 4 G=PSL(2,8) 9 G=PΓL(2,8) 9 G=PΓL(2,32) 33 3 G = M22 22 G = M11 12 G has PSL(2,q) as a minimal normal subgroup q+1 ∗ 3 G=AGL(1,8) 8 G=AΓL(1,8) 8 G=AΓL(1,32) 32 PSL(2,q) ≤ G ≤ PΓL(2,q), q ≡ 3 mod 4 q+1 2 G has one of the following as a minimal normal subgroup (q d −1) PSL(d,q), d ≥ 2 (d,q) 6= (2,2), (2,3) (q−1) (Two representations when d > 2) PSU(3,q),q > 2 q3 + 1 2 2a+1 B2 (q) (Suzuki), q = 2 >2 q2 + 1 2 G2 (q) (Ree) , q = 32a+1 > 3 q3 + 1 2d−1 PSp(2d,2), d > 2 2 ± 2d−1 PSL(2,8) 28 PSL(2,11) (Two representations) 11 A7 (Two representations) 15 HS (Higman-Sims) (Two representations) 176 Co3 (Conway) 276 A = Zp × Zp × · · · × Zp , elementary abelian pn 2∗ G ≤ AΓL(1, q), q ≡ 3 mod 4 q Table 2.2: Finite t-transitive and t∗ -transitive groups
Chapter 3 The Existence of G-semiregular Large Sets The following sections give some examples of the existence and non-existence of Gsemiregular large sets. Section 3.1 settles the existence of [PSL(2, q)]-invariant 3designs, and Section 3.2 gives some examples of the non-existence of [G]-invariant large sets for the Mathieu Groups and a few linear groups.
3.1
PSL(2,q)
Theorem 3.1 At least one k-subset of X is fixed by at least one non-identity element g ∈ PSL(2,q) if and only if one of the following is true:
1. k is even,
2. q ≡ −1 mod d, for some prime d|k,
3. q ≡ 0 mod d, for some prime d|k(k − 1), or 20
21 4. q ≡ 1 mod d, for some prime d|k(k − 1)(k − 2), d 6= 2.
Proof.
Let g ∈ PSL(2,q) have prime order d > 1 and fix some k-subset of X. By
Table 2.1, there are four cases to consider. In each case, Table 2.1 gives the number of fixed points. Case 1: If d = 2, then g fixes no points of X, and consists only of 2-cycles. Thus g fixes some k-subset of X if and only if k is even. Case 2: If d| q+1 and d 6= 2, then g fixes no points of X and consists of only d-cycles. 2 In this case, some k-subset of X is fixed by g if and only if d|k. Also, notice that the condition d| q+1 is equivalent to q ≡ −1 mod d, since d 6= 2 is prime. Thus, g fixes 2 some k-subset of X if and only if q ≡ −1 mod d for some prime d|k. , then g consists of d-cycles and 2 fixed points on X. In this case, Case 3: If d| q−1 2 some k-subset of X is fixed by g if and only if d|k (if g fixes no points of the k-subset), d|(k − 1) (if g fixes one point of the k-subset), or d|(k − 2) (if g fixes two points of the k-subset). Also, notice that the condition d| q−1 is equivalent to q ≡ 1 mod d, since 2 d 6= 2 is prime. Thus, g fixes some k-subset of X if and only if q ≡ 1 mod d for some prime d|k(k − 1)(k − 2). Case 4: If d = p, where q = pe , then g consists of p-cycles and one fixed point on X. In this case, some k-subset of X is fixed by g if and only if d|k (if g fixes no
22 points of the k-subset), or d|(k − 1) (if g fixes one point of the k-subset). Thus g fixes some k-subset of X if and only if p|k(k − 1). Notice that this is equivalent to saying q ≡ 0 mod d for some d|k(k − 1).
Theorem 3.2 Let q = pe ≡ 3 mod 4. Then there exists a [PSL(2,q)]-semiregular LS[N](3,k,q+1), for N =
(q+1 k ) (q+1)q(q−1)/2
, if and only if the following conditions hold
simultaneously.
1. k is odd
2. q 6≡ −1 mod d, for all prime d|k
3. q 6≡ 0 mod d, for all prime d|k(k − 1)
4. q 6≡ 1 mod d, for all prime d|k(k − 1)(k − 2), d 6= 2 (q+1 k ) Proof. Assume there is a [PSL(2,q)]-semiregular LS[N](3,k,q+1) for N= (q+1)q(q−1)/2 . Then by Theorem 1.10 this is equivalent to the fact that no element of PSL(2,q)fixes any k-subset of X. By Lemma 3.1, all of the above conditions must be met.
Theorem 3.3 Let q = pe ≡ 3 mod 4, and q ≥ 2k. Then there exists a PSL(2,q)semiregular LS[N](3,k,q+1) for N = tions hold.
(q+1 k ) (q+1)q(q−1)/2
if and only if the following condi-
23 =1 1. k(k − 1)(k − 2), q−1 2 2. (k(k − 1), p) = 1 3. k, q+1 =1 2 Proof. We will show that this theorem is equivalent to Theorem 3.2. Let d be prime. Then
q+1 ,k 2
=1 ⇔ d6| ⇔
q+1 for all d|k 2
q+1 6≡ 0 mod d for all d|k 2
⇔ q 6≡ −1 mod d for all d|k, and k is odd.
(q, k(k − 1)) = 1 ⇔ d 6 |q, for all d|k(k − 1) ⇔ q 6≡ 0 mod d for all d|k(k − 1).
q−1 , k(k − 1)(k − 2) 2
=1 ⇔ d6| ⇔
q−1 for all d|k(k − 1)(k − 2) 2
q−1 6≡ 0 mod d for all d|k(k − 1)(k − 2) 2
⇔ q 6≡ 1 mod d for all d|k(k − 1)(k − 2), d 6= 2, since q 6≡ 1 mod 2.
24 Lemma 3.1 Let q = pe ≡ 3 mod 4 satisfy the sufficient conditions of 3.2. Then q = 3e or q ≡ 11 mod 12.
Proof.
When q ≡ 3 mod 4, then q ≡ 3, 7 or 11 mod 12. If q ≡ 3 mod 12, then
q=3 + 12s=3(1 + 4s). Since q is a prime power, q = 3e is evident. If q ≡ 7 mod 12, then q ≡ 1 mod 3, but condition 3) from Theorem 3.2 requires q 6≡ 1 mod 3, since clearly 3|k(k − 1)(k − 2). Thus, q 6≡ 7 mod 12. Thus, either q ≡ 11 mod 12, or q = 3e . Lemma 3.2 If q ≡ 11 mod 12 satisfies the sufficient conditions of Theorem 3.2, then k ≡ 1 or 5 mod 6.
Proof.
Notice that condition 1) from Theorem 3.2 requires that q 6≡ 1 mod 2, and
q 6≡ 2 mod 3. But q ≡ 11 mod 12 implies q ≡ 1 mod 2 and q ≡ 2 mod 3.
Note, that since q ≡ 11 mod 12 implies q ≡ 1 mod 2 and q ≡ 2 mod 3, and since k ≡ 1 or 5 mod 6, the primes in Theorem 3.2 can be assumed to be greater than 3.
Theorem 3.4 Let k ≡ 1 or 5 mod 6. Then there are infinitely many primes q which satisfy the sufficient conditions of Theorem 3.2.
Proof. Let q ≡ 11 mod 12. Then the primes 2 and 3 don’t restrict k in Theorem 3.2, Q so can be ignored. Let k(k − 1)(k − 2)=( nr=1 ri ei ) 2s 3t , where the ri > 3 are distinct
25 primes, and set r = r1 r2 · · · rn . Then the equations q 6≡ 0, ±1 mod d, for all d|k(k − 1)(k −2) imply the conditions of Theorem 3.2, so produce large sets. By Corollary 1.1 there are (r1 − 3)(r2 − 3) · · · (rn − 3) solutions modulo r to these equations. Since the ri > 3, this product is non-zero. By Dirichlet’s Theorem, each of these solutions produces infinitely many solutions.
Although Theorem 3.4 gives infinitely many large sets, one might ask if this is a very sparse family. It is easy to write an algorithm to test the conditions of Theorem 3.2 for all congruence classes modulo r. Table 3.1 gives a short list of the number of solutions modulo r, and the density, defined as the number of solutions divided by 4 × φ(r), since the primes fall uniformly into the φ(r) residue classes modulo r, and since we also require that p ≡ 11 mod 12, which constitutes a quarter of the primes. Table A.1 in Appendix A gives a larger table, including k ≤ 223. A few comments are in order. First, notice that for most k in the table, the density is usually somewhere between .10 and .20, which is a significant percentage of the primes. When considering the fact that only one quarter of the primes (those p for which p ≡ 11 mod 12) even have a chance of satisfying the necessary conditions, the density seems even more impressive. Another thing to notice is that this doesn’t include prime powers, for which there are certainly many more large sets. In order
k k(k-1)(k-2) r φ(r) Θ(3.2) Θ(3.4) Density 5 60 5 4 2 2 0.125000 7 210 35 24 12 8 0.125000 11 990 55 40 24 16 0.150000 13 1716 143 120 90 80 0.187500 17 4080 85 64 42 28 0.164063 19 5814 323 288 240 224 0.208333 23 10626 1771 1320 900 640 0.170455 25 13800 115 88 42 40 0.119318 29 21924 203 168 130 104 0.193452 31 26970 4495 3360 2268 1456 0.168750 35 39270 6545 3840 1080 896 0.070313 37 46620 1295 864 510 272 0.147569 41 63960 2665 1920 1254 760 0.163281 43 74046 12341 10080 7800 6080 0.193452 47 97290 5405 4048 2772 1760 0.171196 49 110544 329 276 180 176 0.163044
26
Table 3.1: The density of p for which there exists a PSL(2,p)-semiregular large sets.
to get an estimate of the density of large sets from prime powers, one would need to know the density of prime powers modulo n for a given n, which, to my knowledge, is unknown. When q = 3e , I haven’t shown as strong a result, but a few things of interest can be said. Lemma 3.3 If q = 3e satisfies the sufficient conditions of Theorem 3.2, then k ≡ 5 mod 6. Proof. The sufficient conditions of Theorem 3.2 require that (k(k − 1), 3e ) = 1 and ) = 1 The first equation implies k ≡ 2 mod 3, and the second that k ≡ 1 mod 2 (k, q+1 2
27 since
q+1 2
≡ 0 mod 2. Together, these give k ≡ 5 mod 6.
Theorem 3.5 Let q = 3e , e odd. Then a set of sufficient conditions for the existence of a PSL(2,q)-semiregular LS[N](3,k,q+1), with N =
(q+1 k ) (q+1)q(q−1)/2
, is
1. k(k − 1)(k − 2), q−1 =1 2 =1 2. k, q+1 2 3. k ≡ 5 mod 6
Proof.
Theorem 3.3 shows that the sufficient conditions from Theorem 3.2 imply
those of this theorem. Now, when q = 3e , and k ≡ 5 mod 6, it is clear that (k(k − 1), 3e ) = 1. Thus the conditions above are sufficient.
3.2
t- and t∗-transitive groups with t ≥ 3
When discussing transitive groups in this section, the groups Sn and An are omitted since they are not interesting in the present context. We also concern ourselves only with those groups which are at least 3∗ -transitive. In this section we show that their are no semiregular large sets for t ≥ 4, except perhaps from PΓL(2, 32), with 10 ≤ k ≤ 16, and the only semiregular large sets for t = 3 come from groups with PSL(2,q) as a subgroup, and possibly AΓL(2, 32).
28 If G is a group which is at least 4∗ -transitive, then G is one of M24 , M23 , M12 , M11 , PSL(2,8), PΓL(2, 8), or PΓL(2, 32). If G is 3- or 3∗ -transitive, then G has PSL(2, q) as a subgroup, or G is one of M22 , M11 (acting on 12 points), AGL(1,8), AΓL(1, 8), or AΓL(1, 32). It is not too difficult to determine the existence or non-existence of semiregular large sets from the groups M24 , M23 , M12 , M11 , M22 , M11 (acting on 12 points), PSL(2,8), PΓL(2, 8), PΓL(2, 32), AGL(1,8), AΓL(1, 8), and AΓL(1, 32). One can compute the orbits on k sets and easily determine whether or not the G-orbits have length |G|. In fact, one can also compute the A∆,Γ matrix for the permissible k and determine whether there exist [G]-invariant large sets. The Mathieu Groups yield no semiregular large sets. For M22 , M23 , and M24 , this is clear from [25]. It is easy to compute A∆,Γ for M11 and M12 in their usual representations, and M11 acting 12 points, and come to the same conclusion. Notice that PSL(2, 8) and PΓL(2,8) are 4∗ -transitive, and of degree 9, so they are set-transitive. In other words, there is only 1 orbit of k-subsets for 1 ≤ k ≤ 9, and there are no semiregular large sets from these groups. Computing the A∆,Γ matrices for AGL(1, 8), AΓL(1,8) is very simple, and shows easily that there are no semiregular large sets for these groups.
29 I have verified for 5 ≤ k ≤ 9 that there are no PΓL(2, 32)-semiregular large sets, and I suspect this will also hold for 10 ≤ k ≤ 16 as well. I have not done any computations for AΓL(1,32), although it should not be too difficult. From Table 2.2 and the above discussion, it is clear that their are no semiregular large sets for t ≥ 4, except perhaps from PΓL(2, 32), with 10 ≤ k ≤ 16, and the only semiregular large sets for t = 3 come from groups with PSL(2,q) as a subgroup, and possibly AΓL(2, 32).
Chapter 4 Recursive Constructions for Large Sets 4.1
Known Recursive Constructions
There are several known recursive constructions for large sets. Two are obvious. The first comes from the fact that the existence of a LS[N](t, k, v) implies the existence of a LS[M](t, k, v) for all M dividing N. The second comes from the fact that the existence of an LS[N](t, k, v) implies the existence of a LS[N](t − 1, k − 1, v − 1) by taking the derived design with respect to a particular fixed point x0 of each design in the large set. Several other recursive constructions for large sets of t-designs are known. The following list is taken from [11].
Theorem 4.1 If there exist an LS[M](t, k, v) and an LS[N](t, k + 1, v), then there exists an LS[gcd(M, N)](t, k + 1, v + 1).
30
31 Theorem 4.2 (Teirlinck [41]) For every natural number t let λ(t) = lcm{ lcm{1, 2, . . . , t + 1}, and let ℓ(t) =
Qt
i=1
t m
: m = 1, 2, . . . , t}, let λ∗ (t) =
λ(i) · λ∗ (i). Then, for all N > 0, there is an
LS[N](t, t + 1, t + N · ℓ(t)).
Theorem 4.3 (Khosrovshahi and Ajoodani-Namini [23]) If there are LS[N](t, t + 1, v) and LS[N](t, t + 1, w), then there is also an LS[N](t, t + 1, v + w − t).
Theorem 4.4 (Qiu-rong Wu [42]) If there exist large sets LS[N](t, k, v), LS[N](t, k, w), LS[N](k − 2, k − 1, v − 1), LS[N](k − 2, k − 1, w − 1), then there exists a large set LS[N](t, k, v + w − k + 1).
Corollary 4.1 If there exist large sets LS[N](t, k, v) and LS[N](k − 2, k − 1, v − 1), then there exist large sets LS[N](t, k, v + m(v − k + 1)), for all m ≥ 0.
Theorem 4.5 (Ajoodani-Namini [3] ) If there exists an LS[N](t, m, v − 1), and mN < k < (m + 1)N, then a LS[N](t + 1, k, Nv) also exists.
32
4.2
Applying the Recursive Constructions
Applying the recursive constructions is very elementary, since it only involves looking at the parameters of a large set, not at the actual structures themselves. I have written a program that applies the recursive constructions. Starting with all known large sets having v ≤ 18, of which there are approximately 100, the above constructions yield about 300 new large sets with v ≤ 50. When v is allowed to be as large as 300, these constructions yield about 5600 new large sets. Table 4.1 gives the list of the known large sets with v ≤ 18, plus all large sets with v ≤ 30 which are constructed from these base sets. Table 4.1: Large sets with v ≤ 30 parameters LS[2](2,3,6) LS[2](2,3,10) LS[2](2,3,14) LS[2](2,3,18) LS[2](2,3,22) LS[2](2,3,26) LS[2](2,3,30) LS[2](2,4,10) LS[2](2,4,11) LS[2](2,4,18) LS[2](2,4,19) LS[2](2,4,27) LS[2](2,5,10) LS[2](2,5,11) LS[2](2,5,12) LS[2](2,5,18) LS[2](2,5,19) LS[2](2,5,20) LS[2](2,5,28) LS[2](2,6,12)
λ 2 4 6 8 10 12 14 14 18 60 68 150 28 42 60 280 340 408 1300 105
Method Operands Bhattacharya LS [4](2,3,10) exists 2—4 Hanani LS [8](2,3,18) exists 2—8 KAN LS[2](2,3,6) LS[2](2,3,18) KAN LS[2](2,3,10) LS[2](2,3,18) KAN LS[2](2,3,14) LS[2](2,3,18) LS [14](2,4,10) exists 2—14 LS [6](2,4,11) exists 2—6 LS [10](2,4,18) exists 2—10 Chee LS[2](2,3,18) LS[2](2,4,18) Wu LS[2](2,4,11) LS[2](2,4,19) LS[2](2,3,10) LS[2](2,3,18) LS [14](2,5,10) exists 2—14 LS [42](2,5,11) exists 2—42 LS [6](2,5,12) exists 2—6 Magliveras and Laue Chee LS[2](2,4,18) LS[2](2,5,18) Wu LS[2](2,5,12) LS[2](2,5,12) LS[2](3,4,11) LS[2](3,4,11) Wu LS[2](2,5,12) LS[2](2,5,20) LS[2](3,4,11) LS[2](3,4,19) LS [42](2,6,12) exists 2—42
parameters LS[2](2,6,13) LS[2](2,6,18) LS[2](2,6,19) LS[2](2,6,20) LS[2](2,6,21) LS[2](2,6,29) LS[2](2,7,14) LS[2](2,7,18) LS[2](2,7,19) LS[2](2,7,20) LS[2](2,7,21) LS[2](2,7,22) LS[2](2,7,30) LS[2](2,8,21) LS[2](2,8,22) LS[2](2,8,23) LS[2](2,9,18) LS[2](2,9,20) LS[2](2,9,22) LS[2](2,9,23) LS[2](2,9,24) LS[2](2,10,23) LS[2](2,10,24) LS[2](2,10,25) LS[2](2,11,24) LS[2](2,11,25) LS[2](2,11,26) LS[2](3,4,11) LS[2](3,4,19) LS[2](3,4,27) LS[2](3,5,11) LS[2](3,5,12) LS[2](3,5,20) LS[2](3,5,28) LS[2](3,6,12) LS[2](3,6,13) LS[2](3,6,21) LS[2](3,6,29) LS[2](3,7,14) LS[2](3,7,22) LS[2](3,7,30) LS[2](3,8,21) LS[2](3,9,22) LS[2](3,10,21) LS[2](3,10,23) LS[2](3,11,24)
λ 165 910 1190 1530 1938 8775 396 2184 3094 4284 5814 7752 49140 13566 19380 27132 5720 15912 38760 58140 85272 101745 159885 245157 248710 408595 653752 4 8 12 14 18 68 150 42 60 408 1300 165 1938 8775 4284 13566 15912 38760 101745
Method Operands LS [6](2,6,13) exists 2—6 Magliveras and Laue Chee LS[2](2,5,18) LS[2](2,6,18) Chee LS[2](2,5,19) LS[2](2,6,19) Wu LS[2](2,6,13) LS[2](2,6,13) LS[2](4,5,12) Wu LS[2](2,6,13) LS[2](2,6,21) LS[2](4,5,12) LS [12](2,7,14) exists 2—12 LS [8](2,7,18) exists 2—8 Chee LS[2](2,6,18) LS[2](2,7,18) Chee LS[2](2,6,19) LS[2](2,7,19) Chee LS[2](2,6,20) LS[2](2,7,20) Wu LS[2](2,7,14) LS[2](2,7,14) LS[2](5,6,13) Wu LS[2](2,7,14) LS[2](2,7,22) LS[2](5,6,13) DER LS[2](3,9,22) Chee LS[2](2,7,21) LS[2](2,8,21) Chee LS[2](2,7,22) LS[2](2,8,22) LS [10](2,9,18) exists 2—10 DER LS[2](3,10,21) DER LS[2](3,10,23) Chee LS[2](2,8,22) LS[2](2,9,22) Chee LS[2](2,8,23) LS[2](2,9,23) DER LS[2](3,11,24) Chee LS[2](2,9,23) LS[2](2,10,23) Chee LS[2](2,9,24) LS[2](2,10,24) DER LS[2](3,12,25) Chee LS[2](2,10,24) LS[2](2,11,24) Chee LS[2](2,10,25) LS[2](2,11,25) derivation of LS[4](4,5,12 KAN LS[2](3,4,11) LS[2](3,4,11) Wu LS[2](3,4,11) LS[2](3,4,19) LS[2](2,3,10) Magliveras and Laue LS [6](3,5,12) exists 2—6 Wu LS[2](3,5,12) LS[2](3,5,12) LS[2](3,4,11) Wu LS[2](3,5,12) LS[2](3,5,20) LS[2](3,4,11) LS [42](3,6,12) exists 2—42 LS [6](3,6,13) exists 2—6 Wu LS[2](3,6,13) LS[2](3,6,13) LS[2](4,5,12) Wu LS[2](3,6,13) LS[2](3,6,21) LS[2](4,5,12) Magliveras and Laue Wu LS[2](3,7,14) LS[2](3,7,14) LS[2](5,6,13) Wu LS[2](3,7,14) LS[2](3,7,22) LS[2](5,6,13) DER LS[2](4,9,22) DER LS[2](4,10,23) DER LS[2](4,11,22) DER LS[2](4,11,24) DER LS[2](4,12,25)
33
LS[2](4,5,12) LS[2](4,5,20)
LS[2](5,6,13) LS[2](5,6,21)
LS[2](2,3,18)
LS[2](3,4,11) LS[2](3,4,19)
LS[2](4,5,12) LS[2](4,5,20) LS[2](5,6,13) LS[2](5,6,21)
parameters LS[2](3,12,25) LS[2](4,5,12) LS[2](4,5,20) LS[2](4,5,28) LS[2](4,6,12) LS[2](4,6,13) LS[2](4,6,21) LS[2](4,6,29) LS[2](4,7,14) LS[2](4,7,22) LS[2](4,7,30) LS[2](4,9,22) LS[2](4,10,23) LS[2](4,11,22) LS[2](4,11,24) LS[2](4,12,25) LS[2](4,13,26) LS[2](5,6,13) LS[2](5,6,21) LS[2](5,6,29) LS[2](5,7,14) LS[2](5,7,22) LS[2](5,7,30) LS[2](5,11,24) LS[2](5,12,25) LS[2](5,13,26) LS[2](6,7,14) LS[2](6,7,22) LS[2](6,7,30) LS[2](6,13,26) LS[3](2,3,11) LS[3](2,3,20) LS[3](2,3,29) LS[3](2,4,11) LS[3](2,4,12) LS[3](2,4,21) LS[3](2,4,30) LS[3](2,5,11) LS[3](2,5,12) LS[3](2,5,13) LS[3](2,5,22) LS[3](2,6,12) LS[3](2,6,13) LS[3](2,6,14) LS[3](2,6,23) LS[3](2,7,14)
λ 248710 4 8 12 14 18 68 150 60 408 1300 4284 13566 15912 38760 101745 248710 4 8 12 18 68 150 13566 38760 101745 4 8 12 38760 3 6 9 12 15 57 126 28 40 55 380 70 110 165 1995 264
Method Operands DER LS[2](4,13,26) Denniston KAN LS[2](4,5,12) LS[2](4,5,12) Wu LS[2](4,5,12) LS[2](4,5,20) Magliveras and Laue Magliveras and Laue Wu LS[2](4,6,13) LS[2](4,6,13) Wu LS[2](4,6,13) LS[2](4,6,21) Magliveras and Laue Wu LS[2](4,7,14) LS[2](4,7,14) Wu LS[2](4,7,14) LS[2](4,7,22) AN LS[2](3,4,11) DER LS[2](5,11,24) AN LS[2](3,5,11) AN LS[2](3,5,12) DER LS[2](5,13,26) AN LS[2](3,6,13) derivation of LS[4](6,7,14 KAN LS[2](5,6,13) LS[2](5,6,13) Wu LS[2](5,6,13) LS[2](5,6,21) Magliveras and Laue Wu LS[2](5,7,14) LS[2](5,7,14) Wu LS[2](5,7,14) LS[2](5,7,22) AN LS[2](4,5,12) DER LS[2](6,13,26) AN LS[2](4,6,13) Kreher and Radziszowski KAN LS[2](6,7,14) LS[2](6,7,14) Wu LS[2](6,7,14) LS[2](6,7,22) AN LS[2](5,6,13) Teirlinck KAN LS[3](2,3,11) LS[3](2,3,11) DER LS[3](3,4,30) LS [6](2,4,11) exists 3—6 LS [15](2,4,12) exists 3—15 Wu LS[3](2,4,12) LS[3](2,4,12) Wu LS[3](2,4,12) LS[3](2,4,21) LS [42](2,5,11) exists 3—42 LS [6](2,5,12) exists 3—6 Magliveras and Laue Wu LS[3](2,5,13) LS[3](2,5,13) LS [42](2,6,12) exists 3—42 LS [6](2,6,13) exists 3—6 Magliveras and Laue Wu LS[3](2,6,14) LS[3](2,6,14) LS [12](2,7,14) exists 3—12
34
LS[2](3,4,11) LS[2](3,4,19)
LS[2](4,5,12) LS[2](4,5,12) LS[2](4,5,12) LS[2](4,5,20) LS[2](5,6,13) LS[2](5,6,13) LS[2](5,6,13) LS[2](5,6,21)
LS[2](4,5,12) LS[2](4,5,20) LS[2](5,6,13) LS[2](5,6,13) LS[2](5,6,13) LS[2](5,6,21)
LS[2](5,6,13) LS[2](5,6,21)
LS[3](2,3,11) LS[3](2,3,11) LS[3](2,3,11) LS[3](2,3,20)
LS[3](3,4,12) LS[3](3,4,12)
LS[3](4,5,13) LS[3](4,5,13)
parameters LS[3](2,7,15) LS[3](3,4,12) LS[3](3,4,21) LS[3](3,4,30) LS[3](3,5,12) LS[3](3,5,13) LS[3](3,5,22) LS[3](3,6,12) LS[3](3,6,13) LS[3](3,6,14) LS[3](3,6,23) LS[3](3,7,14) LS[3](3,7,15) LS[3](4,5,13) LS[3](4,5,22) LS[3](4,6,13) LS[3](4,6,14) LS[3](4,6,23) LS[3](4,7,14) LS[3](4,7,15) LS[4](2,3,10) LS[4](2,3,18) LS[4](2,3,26) LS[4](2,7,14) LS[4](2,7,18) LS[5](2,3,12) LS[5](2,3,22) LS[5](2,4,8) LS[5](2,4,12) LS[5](2,4,13) LS[5](2,4,18) LS[5](2,4,23) LS[5](2,9,18) LS[5](3,4,13) LS[5](3,4,23) LS[6](2,4,11) LS[6](2,5,11) LS[6](2,5,12) LS[6](2,6,12) LS[6](2,6,13) LS[6](2,7,14) LS[6](3,5,12) LS[6](3,6,12) LS[6](3,6,13) LS[7](2,3,9) LS[7](2,3,16)
λ 429 3 6 9 12 15 57 28 40 55 380 110 165 3 6 12 15 57 40 55 2 4 6 198 1092 2 4 3 9 11 24 42 2288 2 4 6 14 20 35 55 132 6 14 20 1 2
Method Operands Chee LS[3](2,6,14) LS[3](2,7,14) Teirlinck KAN LS[3](3,4,12) LS[3](3,4,12) Wu LS[3](3,4,12) LS[3](3,4,21) LS [6](3,5,12) exists 3—6 Chee, Colbourn, Furino, Kreher Wu LS[3](3,5,13) LS[3](3,5,13) LS [42](3,6,12) exists 3—42 LS [6](3,6,13) exists 3—6 Magliveras and Laue Wu LS[3](3,6,14) LS[3](3,6,14) Magliveras and Laue Chee LS[3](3,6,14) LS[3](3,7,14) Kramer, Magliveras and O’Brien KAN LS[3](4,5,13) LS[3](4,5,13) Magliveras and Laue Chee, Colbourn, Furino, Kreher Wu LS[3](4,6,14) LS[3](4,6,14) Magliveras and Laue Chee LS[3](4,6,14) LS[3](4,7,14) Teirlinck LS [8](2,3,18) exists 4—8 KAN LS[4](2,3,10) LS[4](2,3,18) LS [12](2,7,18) exists 4—12 LS [8](2,7,18) exists 4—8 Schreiber KAN LS[5](2,3,12) LS[5](2,3,12) Sharry and Street LS [15](2,4,12) exists 5—15 LS [55](2,4,13) exists 5—55 LS [10](2,4,18) exists 5—10 Wu LS[5](2,4,13) LS[5](2,4,13) LS [10](2,9,18) exists 5—10 Kramer, Magliveras and O’Brien KAN LS[5](3,4,13) LS[5](3,4,13) Chee, Colbourn, Furino, Kreher LS [42](2,5,11) exists 6—42 LS[6](3,5,12 as 2- designs LS [42](2,6,12) exists 6—42 Chee, Magliveras LS [12](2,7,14) exists 6—12 Kramer, Magliveras and Stinson LS [42](3,6,12) exists 6—42 Chee, Magliveras Kirkman Magliveras and Laue
35
LS[3](2,3,11) LS[3](2,3,20)
LS[3](3,4,12) LS[3](3,4,12)
LS[3](4,5,13) LS[3](4,5,13)
LS[3](4,5,13) LS[3](4,5,13)
LS[5](2,3,12) LS[5](2,3,12)
parameters LS[7](2,3,23) LS[7](2,3,30) LS[7](2,4,9) LS[7](2,4,10) LS[7](2,4,17) LS[7](2,4,24) LS[7](2,5,10) LS[7](2,5,11) LS[7](2,5,18) LS[7](2,6,12) LS[7](3,4,17) LS[7](3,5,10) LS[7](3,6,12) LS[11](2,3,13) LS[11](2,3,24) LS[11](2,4,13) LS[11](2,4,14) LS[11](2,4,25) LS[11](2,5,13) LS[11](2,5,14) LS[11](2,5,15) LS[11](2,6,13) LS[11](2,6,14) LS[11](2,6,15) LS[11](2,6,16) LS[13](2,3,15) LS[13](2,3,28) LS[14](2,4,10) LS[14](2,5,10) LS[14](2,5,11) LS[14](2,6,12) LS[14](3,6,12) LS[15](2,4,12) LS[21](2,5,11) LS[21](2,6,12) LS[21](3,6,12) LS[42](2,5,11) LS[42](2,6,12) LS[42](3,6,12) LS[55](2,4,13)
λ 3 4 3 4 15 33 8 12 80 30 2 3 12 1 2 5 6 23 15 20 26 30 45 65 91 1 2 2 4 6 15 6 3 4 10 4 2 5 2 1
Method Operands KAN LS[7](2,3,9) LS[7](2,3,16) KAN LS[7](2,3,16) LS[7](2,3,16) Kramer, Magliveras and Stinson LS [14](2,4,10) exists 7—14 Wu LS[7](2,4,10) LS[7](2,4,10) LS[7](2,3,9) LS[7](2,3,9) Wu LS[7](2,4,10) LS[7](2,4,17) LS[7](2,3,9) LS[7](2,3,16) LS [14](2,5,10) exists 7—14 LS [42](2,5,11) exists 7—42 Magliveras and Laue LS [14](2,6,12) exists 7—42 Chee, Magliveras extension of LS[7](2,4,9 LS [42](3,6,12) exists 7—42 Denniston KAN LS[11](2,3,13) LS[11](2,3,13) LS [55](2,4,13) exists 11—55 Kramer, Magliveras and O’Brien Wu LS[11](2,4,14) LS[11](2,4,14) LS[11](2,3,13) LS[11](2,3,13) Chee, Magliveras Chee, Magliveras Chee, Magliveras Chee, Magliveras Chee LS[11](2,5,13) LS[11](2,6,13) Chee LS[11](2,5,14) LS[11](2,6,14) Chee LS[11](2,5,15) LS[11](2,6,15) Denniston KAN LS[13](2,3,15) LS[13](2,3,15) Kramer, Magliveras and Stinson Kramer, Magliveras and Stinson LS [42](2,5,11) exists 14—42 LS [42](2,6,12) exists 14—42 DIV LS[42](3,6,12) Kramer, Magliveras and Stinson LS [42](2,5,11) exists 21—42 LS [42](2,6,12) exists 21—42 LS [42](3,6,12) exists 21—42 Kramer, Magliveras and Stinson LS [42](3,6,12 as 2- designs extension of LS [42](2,5,11 Chouinard
36
Chapter 5 Further Research There are many areas for further research in large sets, particularly related to several of the topics of this thesis. In terms of PSL(2,q) being an automorphism group of a large set, what more can be said? We have settled the question for PSL(2,q)semiregular large sets, but there may be other [PSL(2, q)]-invariant or PSL(2, q)invariant large sets. For example, the existence of [PSL(2, q)]-invariant large sets for k = 4 and k = 5 has been settled [14]. It should be noted that the [PSL(2, q)]-invariant large sets for k = 5 are actually PSL(2,q)-semiregular. One might ask whether there is a general result for [PSL(2, q)]-invariant large sets with arbitrary k. In finding G-semiregular large sets, one only considers those matrices A∆,Γ which have one row. It has been suggested by others that it may be worth while to consider the case when A∆,Γ has two or three rows. As shown in Table 2.2, there are many t- and t∗ - transitive groups which may yield 37
38 G-semiregular or [G]-invariant large sets. For t ≥ 3, the only groups which have not been considered extensively are PΓL(2,32), AΓL(1,32), and groups with PSL(2, q) as a subgroup. It seems unlikely that there are PΓL(2, 32)-semiregular large sets, although there may be [P ΓL(2, 32)]-invariant large sets. I suspect the same is true of AΓL(1,32). With a little work, the groups PΓL(2,32) and AΓL(1,32) can be settled, leaving the groups with PSL(2, q) as a subgroup to consider. There are also many 2and 2∗ -transitive groups to consider. Magliveras and Laue, among others, have proposed a large database containing the known large sets. The database I have created is very modest in terms of capability and size. To be complete, one should add to the database some of the other known large sets, including several infinite families. In addition, other recursive constructions for large sets should be investigated. In particular, none of the known constructions increases the parameter N. Thus, most of the constructions increase λ, making the chance of finding a large set with minimal λ using a construction small. Although I am skeptical that such a construction exists, it certainly is worth considering.
Appendix A Density of Large Sets from PSL(2,q) Table A.1 gives the density of primes q for which there exist PSL(2,q)-semiregular large sets, for k ≤ 223. r is the product of prime divisors of k(k − 1)(k − 2) greater than 3, Θ(3.2) and Θ(3.4) are the number of solutions modulo r due to Theorems 3.2 and 3.4, respectively, and Density is Θ(3.2)/(4 × φ(r)).
39
40 Table A.1: The density of p for which there exists a PSL(2,p)-semiregular large sets. k k(k-1)(k-2) r φ(r) Θ(3.2) Θ(3.4) Density 5 60 5 4 2 2 0.125000 7 210 35 24 12 8 0.125000 11 990 55 40 24 16 0.150000 13 1716 143 120 90 80 0.187500 17 4080 85 64 42 28 0.164063 19 5814 323 288 240 224 0.208333 23 10626 1771 1320 900 640 0.170455 25 13800 115 88 42 40 0.119318 29 21924 203 168 130 104 0.193452 31 26970 4495 3360 2268 1456 0.168750 35 39270 6545 3840 1080 896 0.070313 37 46620 1295 864 510 272 0.147569 41 63960 2665 1920 1254 760 0.163281 43 74046 12341 10080 7800 6080 0.193452 47 97290 5405 4048 2772 1760 0.171196 49 110544 329 276 180 176 0.163044 53 140556 11713 9984 8250 7000 0.206581 55 157410 2915 2080 816 800 0.098077 59 195054 32509 29232 25704 23296 0.219828 61 215940 17995 13920 9918 6496 0.178125 65 262080 455 288 100 80 0.086805 67 287430 47905 31680 19008 10240 0.150000 71 342930 57155 36960 21420 10880 0.144886 73 373176 5183 5040 4830 4760 0.239583 77 438900 7315 4320 1632 1024 0.094445 79 474474 79079 56160 37620 24320 0.167468 83 551286 3403 3280 3120 3040 0.237805 85 592620 49385 31488 11340 8960 0.090034 89 681384 28391 24640 20898 17888 0.212033 91 728910 40495 25344 10440 6880 0.102983 95 830490 138415 99360 41760 39424 0.105072 97 884640 9215 6912 4794 3008 0.173394 101 999900 5555 4000 2646 1568 0.165375 103 1061106 176851 163200 148500 137200 0.227482 107 1190910 198485 132288 79560 41600 0.150354 109 1259604 11663 11448 11130 11024 0.243056
k k(k-1)(k-2) r φ(r) Θ(3.2) Θ(3.4) 113 1404816 29267 24192 19250 14960 115 1481430 246905 177408 75480 70400 119 1642914 91273 66816 35112 31360 121 1727880 6545 3840 1800 896 125 1906500 6355 4800 2262 2128 127 2000250 4445 3024 1860 992 131 2196870 366145 262080 173184 102400 133 2299836 191653 140400 74304 65536 137 2515320 11645 8704 6030 3752 139 2627934 437989 412896 385560 364480 143 2863146 477191 386400 248400 239360 145 2985840 20735 13440 5148 4160 149 3241644 38591 31968 25550 19856 151 3374850 112495 88800 65268 43216 155 3652110 202895 115200 37800 25088 157 3796260 316355 224640 147378 86240 161 4095840 42665 27456 12240 8000 163 4251366 26243 21384 16800 12800 167 4574130 762355 544480 358668 209920 169 4741464 15197 11952 8250 6560 173 5088276 141341 130032 118490 108800 175 5267850 175595 115584 36936 35360 179 5639574 939929 908512 872784 847616 181 5831820 161995 128160 94518 62656 185 6229320 259555 190080 84252 78880 187 6434670 1072445 691200 341040 213248 191 6858810 127015 82080 47940 24064 193 7077696 36863 36480 35910 35720 197 7529340 89635 56448 32010 15520 199 7762194 431233 388080 343980 304192 203 8242206 1373701 1108800 669240 652288 205 8489460 707455 430080 153900 110656 209 8998704 62491 47520 29568 25600 211 9260790 1543465 907200 477360 212992 215 9800130 1633355 1246560 579600 565760 217 10077480 46655 30240 13776 8960 221 10647780 887315 552960 268380 156800 223 10940826 1823471 1534464 1270500 1047200
Density 0.198930 0.106365 0.131376 0.117188 0.117813 0.153770 0.165202 0.132308 0.173196 0.233449 0.160714 0.095759 0.199809 0.183750 0.082031 0.164016 0.111451 0.196409 0.164684 0.172565 0.227809 0.079890 0.240169 0.184375 0.110811 0.123351 0.146016 0.246094 0.141768 0.221591 0.150893 0.089460 0.155556 0.131548 0.116240 0.113889 0.121338 0.206994
41
42 k
k(k-1)(k-2) r φ(r) Θ(3.2) Θ(3.4) Density
Bibliography [1] W.O. Alltop, An infinite class of 5-designs, J. Combin. Theory A, 12 (1972), 390-395. [2] T.M. Apostol, Introduction to Analytic Number Theory, Undergraduate Text in Mathematics, Springer-Verlag, New York, 1976. [3] S. Ajoodani-Namini, Extending Large Sets of t-Designs. J. Combin. Theory A, 76 (1996), 139–144. [4] T. Beth, D Jungnickel and H. Lenz, Design Theory, Cambridge University Press, London, 1986. [5] N. Biggs, Finite Groups of Automorphisms, Cambridge University Press, Oxford, 1971. [6] P.J. Cameron, Finite Permutation Groups and Finite Simple Groups, Bulletin of the London Math. Society, (13) Part 1, Number 40, 1-22 (1981). [7] A.R. Camina, A survey of the Automorphism Groups of Block Designs, Journal of Combinatorial Designs 2 (1994), 79-100. [8] R.W. Carter, Simple Groups of Lie Type, Wiley-Interscience, New York, 1972. [9] Y.M. Chee, C.J. Colbourn and D.L. Kreher, Simple t-designs with v ≤ 30, Ars. Combin. 29 (1990), 193-258. [10] Y.M. Chee, C.J. Colbourn, S.C. Furino and D.L. Kreher, Large Sets of Disjoint t-Designs, Austral. J. Combin. 2 (1990), 111-120. [11] Y.M. Chee and S.M. Magliveras, A Few More Large Sets of t-Designs, To appear in Journal of Combinatorial Designs. [12] S.S. Magliveras, Leo Chouinard and Robert Jajcay, Finite Groups and Designs, CRC Handbook of Combinatorial Designs, CRC Press, C.J. Colbourn and J.H. Dinitz, eds.,(1996), pp 587-615. 43
44 [13] C.A. Cusack, PSL(2, q) As An Automorphism Group Of A 3 − (q + 1, 5, λ) Design Master Thesis, Michigan Technological University, (1994). [14] C.A. Cusack, S.W. Graham, D.L. Kreher, Large Sets of 3-Designs from PSL(2, q) with Block Sizes 4 and 5, Journal of Combinatorial Designs (A), 3 (1995) No 2 147-160. [15] L.E. Dickson, Linear Groups, with an Exposition of the Galois Field Theory, Dover Publications, 1958. [16] J.H. Dinitz and D.R. Stinson (editors), Contemporary Design Theory: A Collection of Surveys, John Wiley & Sons, 1992. [17] D. Gorenstein, Finite Groups, Harper and Row Publishers, 1968. [18] D. Gorenstein, Finite Simple Groups, An Introduction to their Classification, Plenum Press, 1982. [19] W.M. Kantor, 4-Homogeneous Groups, Math. Z., 103 (1968), 67-68. Correction Math. Z. 109 (1969), 86. [20] W.M. Kantor, Automorphism Groups of Designs Math. Z., 109 (1969), 246252. [21] W.M. Kantor, k-Homogeneous Groups, Math. Z., 124 (1972), 261-265. [22] W.M. Kantor, Homogeneous Designs and Geometric Lattices Journal of Combinatorial Theory, Series A 38 66-74 (1985). [23] G.B. Khosrovshahi and S. Ajoodani-Namini, Combining t-Designs, J. Combin. Theory (A) 58 (1991), 26-34. [24] E.S. Kramer, S.S. Magliveras and D.W. Leavitt, Construction Procedures for t-Designs and the Existence of New Simple 6-designs, Annals of Discrete Mathematics, 26 (1985), pp 247-274. [25] E.S. Kramer, S.S. Magliveras, D.M. Mesner, t-designs from the Large Mathieu Groups, Discrete Mathematics 36 (1981), 171-189. [26] E.S. Kramer, S.S. Magliveras, E.A. O’Brien, Some new large sets of t-designs, Austral. J. Combin. 7 (1993), 189-193. [27] E.S. Kramer, S.S. Magliveras, D.R. Stinson, Some small large sets of t-designs, Austral. J. Combin. 3 (1991), 191-205. [28] E.S. Kramer and D.M. Mesner, Intersections among Steiner systems, J. Combin. Theory (A) 16 (1974), 273-285.
45 [29] E.S. Kramer and D.M. Mesner, t-designs on hypergraphs, Discrete Math. 15 (1976), 263-269. [30] D.L. Kreher, Simple t-Designs with large t: A survey, to appear in JCMCC. [31] D. Livingstone, A. Wagner, Transitivity of finite permutation groups of unordered sets, Math. Zeitschr. 90, 393-403 (1965). [32] S.S. Magliveras and D.W. Leavitt, Simple 6-(33,8,36) Designs from P ΓL(2, 32), Proc. of the Durham Computational Group Theory Symposium, Academic Press, (1983), pp 337-352. [33] S.S. Magliveras and D.W. Leavitt, Simple Six-Designs Exist!, Congr. Numer., 40 (1983), pp 195-205. [34] I. Niven, H.S. Zuckerman, and H.L. Montgomery, An Introduction to the Theory of Numbers, Fifth Edition, John Wiley and Sons, Inc., 1991. [35] Joseph J. Rotman, An Introduction to the Theory of Groups, Fourth Edition, Springer Verlag, 1995. [36] L. Teirlinck, On the maximum number of disjoint Steiner triple systems, Discr. Math. 6 (1973), 299-300. [37] L. Teirlinck, On the maximum number of disjoint triple systems, J. Geometry 12 (1975), 93-96. [38] L. Teirlinck, On large sets of disjoint quadruple systems, Ars. Combin. 17 (1984), 173-176. [39] L. Teirlinck, A completion of Lu’s determination of the spectrum for large sets of disjoint Steiner triple systems, J. Combin. Theory (A) 57 (1991), 302-305. [40] L. Teirlinck, Non-trivial t-designs exist for all t, Discr. Math. 65 (1987), 345356. [41] L. Teirlinck, Locally trivial t-designs and t-designs without repeated blocks, Discr. Math. 77 (1989), 345-356. [42] Qiu-rong Wu, A note on Extending t-Designs, Austral. J. Combin. 4 (1991) 229-235.
By Charles A. Cusack
A THESIS
Presented to the Faculty of The Graduate College at the University of Nebraska In Partial Fulfillment of Requirements For the Degree of Master of Science Major: Computer Science
Under the Supervision of Professor Spyros Magliveras
Lincoln, Nebraska
May, 1998
Acknowledgments I would like to start by thanking the faculty of the Department of Computer Science and Engineering and the Department of Mathematics and Statistics at UNL. I have learned many things in the last several years that have helped me in several areas of this thesis. Along with the faculty, I would like to thank the office staff who do much in the department that is unseen, and have certainly done much for me personally. I would also like to thank Phil Romig for teaching me too many things to mention. Also, I would like to thank the members of my committee, Professors Spyros Magliveras, Douglas Stinson, and Jean-Camille Birget. They have taken time out of their busy schedules to read and comment on drafts of the thesis. I would like to thank my adviser, Professor Spyros Magliveras. He has taught me many things in the last 3 years, and has sparked my interest in several areas of research. He encourages me both personally and professionally. Lastly, I would like to thank my Lord and Savior, Jesus Christ, who has been the constant in my life as a student, and will continue to be when I am no longer a student. He has revealed to me a another part of His Big Book of Theorems. He gives meaning and purpose to my life and work, and has saved me from that which I could not save myself. To God be the glory!
Semiregular Large Sets Charles A. Cusack, M.S. University of Nebraska, 1998
Adviser: Spyros Magliveras
A t − (v, k, λ) design (X, B) is a v-element set X of points and a collection B of kelement subsets of X called blocks such that every t-element subset of X is contained in precisely λ blocks. The collection design with λ=λ=
v−t k−t
X k
of all k-subsets of a set X forms a t-(v,k,λ)
, and is called the complete design.
An LS[N](t, k, v), or a large sets of t − (v, k, λ) designs, is a partition, [(X, Bi )]N i=1 , of the complete design into N disjoint t − (v, k, λ/N) designs. These are also called large sets of t − (v, k, λ) designs. A group G is said to be an automorphism group of a large set B if Bg =B for all g ∈ G. That is, if Big ∈ B for all Bi ∈ B and g ∈ G. Equivalently, we say that a large set with this property is G-invariant. If the stronger condition that Big =Bi for all Bi ∈ B and g ∈ G holds, we say that a large set is [G]-invariant. If the G-orbits of a set X are all of length |G|, we say that G acts semiregularly on X. In particular, if t < k, G is t-homogeneous on X, and G acts semiregularly
on the k-subsets of X, then there exists an LS[N](t, k, v). In this case, we say it is a G-semiregular large set. In chapter 2, we discuss transitive groups and give a few details about the group PSL(2, q). Then in chapter 3 we discuss the existence of PSL(2, q)-semiregular LS[N] (3,k,q+1). In chapter 4 we discuss the use of recursive constructions to find new large sets. In particular, we give the parameters for some large sets with small v that can be obtained using these constructions.
Contents
1 Introduction
1
1.1
Set and Number Theory . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.2
Design Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.3
Group Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.4
Large Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
2 Transitive Groups
14
2.1
Examples of transitive groups . . . . . . . . . . . . . . . . . . . . . .
14
2.2
A list of all t- and t∗ -transitive groups . . . . . . . . . . . . . . . . . .
16
2.3
PSL(2,q) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
3 The Existence of G-semiregular Large Sets
20
3.1
PSL(2,q) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
3.2
t- and t∗ -transitive groups with t ≥ 3 . . . . . . . . . . . . . . . . . .
27
i
4 Recursive Constructions for Large Sets
30
4.1
Known Recursive Constructions . . . . . . . . . . . . . . . . . . . . .
30
4.2
Applying the Recursive Constructions . . . . . . . . . . . . . . . . . .
32
5 Further Research
37
A Density of Large Sets from PSL(2,q)
39
ii
List of Figures 1.1
A graphical representation of a 2 − (7, 3, 1) design . . . . . . . . . . .
iii
5
List of Tables 2.1
Useful facts about PSL(2,q) when q ≡ 3 mod 4 . . . . . . . . . . . . .
18
2.2
Finite t-transitive and t∗ -transitive groups . . . . . . . . . . . . . . .
19
3.1
The density of p for which there exists a PSL(2,p)-semiregular large sets. 26
4.1
Large sets with v ≤ 30 . . . . . . . . . . . . . . . . . . . . . . . . . .
32
A.1 The density of p for which there exists a PSL(2,p)-semiregular large sets. 40
iv
Chapter 1 Introduction The theory of t − (v, k, λ) designs, or simply t−designs, is an important part of combinatorics. The question of existence of non-trivial t−designs for t > 5 was open until the early 1980’s. Motivated by the classification of finite simple groups, and the corollary that t−transitive groups do not exist for t > 5, some researchers thought that for t > 5, t-designs could not exist. This conjecture was proved wrong by Leavitt and Magliveras in 1983 [32, 33] when they constructed the first simple 6-designs from the exceptional 4-homogeneous group P ΓL2 (32) acting on the projective line of 33 points. Infinite families of 5-designs were not known to exist until 1972 [1]. In fact, with the exception of two 5-designs found by E. Kramer, all known 5-designs found before 1982 were on an even number of points. The combinatorial design research landscape changed dramatically in 1985 when L. Teirlinck [40] showed that simple t−designs exist for arbitrarily large t and v. It became obvious from Teirlinck’s 1
2 construction that the “high-road” to the construction of t−designs for large t is by means of the notion of Large Sets of t−designs. A large set constitutes a partition of the collection of all k−subsets of a given point set X into disjoint t − (v, k, λ) designs. In the last dozen years, several recursive constructions resulting in infinite families of large sets of t−designs have been discovered, and we think there are certainly more to come. These constructions have as corollaries the existence of t−designs with both small and large t, previously unknown to exist. To date, no large sets of t−designs are known to exist with t > 5 and λ = 1 (Steiner systems.) In fact, not even a single non-trivial t−design is known for t > 5 with λ = 1, let alone a large set of such designs. Further, even though infinitely many large sets of 3-designs are known, again, not a single large set of 3-(v,4,1) designs is known.
1.1
Set and Number Theory
Let X be a set. If X consists of n points, we say X has cardinality n, and denote this by |X|=n. If K ⊆ X has cardinality k, we sat that K is a k-subset of X. We denote the set of all k-subsets of a set X by
X k
. It is not hard to see that
X k
=
|X| k
.
Let x, y, and z be integers. We write x ≡ y mod z when x=y+wz for some integer w. The following theorem is well known.
3 Theorem 1.1 Chinese Remainder Theorem
Let m1 , m2 , . . . , mr denote r
positive integers that are pairwise relatively prime, let m = m1 m2 · · · mr , and let a1 , a2 , . . . , ar denote any r integers with 0 ≤ ai ≤ mi − 1. Then the system of r congruences x ≡ ai mod mi (1 ≤ i ≤ r) has a unique solution modulo m.
The following is a simple extension of the Chinese Remainder Theorem.
Corollary 1.1 Let r1 , r2 , . . . , rn be pairwise relatively prime positive integers, with r = r1 r2 · · · rn , and let t1 , t2 , . . . , tn be positive integers such that 0 ≤ ti ≤ ri for 1 ≤ i ≤ n. For each i = 1, . . . , n, let si1 , si2 , . . . , siti be distinct integers modulo ri . Then the set of relations
x ∈ {s11 , s12 , . . . , s1t1 } mod r1 , and x ∈ {s2 , s2 , . . . , s2 } mod r2 , and t2 1 2 .. . x ∈ {s , s , . . . , s } mod r n1 n2 nt n n has exactly t1 t2 · · · t3 distinct solutions modulo r.
The next two theorems relate to the distribution of primes.
Theorem 1.2 (Dirichlet) If gcd(a, k) = 1, then there are infinitely many primes p ≡ a mod k.
4 Theorem 1.3 The primes are uniformly distributed among the φ(k) residue classes modulo k that correspond to invertible elements in Zk .
For more information about number theory, see [34]. For an in-depth treatment of the distribution of primes, see chapter 7 of [2].
1.2
Design Theory
A simple t − (v, k, λ) design, (X, B), is a v-element set X of points and a collection B of k-element subsets of X called blocks, such that every t-element subset of X is contained in precisely λ blocks. All designs in this thesis will be simple. It is well known that for each 0 ≤ s ≤ t every t − (v, k, λ) design is also an s − (v, k, λs ) design, where k−s v−s . λs = λ t−s t−s Thus, a set of necessary conditions (called the divisibility conditions) for the existence of a t − (v, k, λ) design is that
v−s λ t−s
k−s t−s
be an integer for s = 0, 1, 2, . . . , t − 1. We denote by λ∗ (t, k, v) (or simply λ∗ , if the parameter set is clear) the minimum positive integer λ that satisfies the divisibility
5 conditions. The number of blocks in the design is k v . b = λ0 = λ t t Example: A 2 − (7, 3, 1) design is given by X = {0, 1, 2, 3, 4, 5, 6}, and B={013, 124, 235, 346, 450, 561, 602}. Figure 1.1 shows a graphical representation of the design. The blocks are the 6 lines and 1 circle. This design is the smallest projective plane, known as the Fano plane. 0
1
5 2
3
6
4
Figure 1.1: A graphical representation of a 2 − (7, 3, 1) design
For a thorough treatment of design theory, see [16].
1.3
Group Theory
For basic definitions in group theory, see [35]. We denote by SX the symmetric group on the set X. Let X be a set, let G be a group, and suppose the formal multiplication
X ×G→X
(x, g) 7→ xg
6 is defined and satisfies the following axioms:
1. x1 = x for all x ∈ X, where 1 is the identity in G. 2. (x)gh = (xg )h for all x ∈ X, and for all g, h ∈ G.
Then we say that G acts on the set X, and we write G|X. If H ⊆ G, we define xH ={xh : h ∈ H}. The G-orbit of x, or simply the orbit of x, is the set xG . The stabilizer of x ∈ X in G is Gx = {g ∈ G : xg = x}. It is not hard to see that Gx is a subgroup of G. A group action G|X is called semiregular if all G-orbits of X are of length |G|. A group action G|X is called transitive if X consists of one G-orbit (i.e., xG =X for all x ∈ X). Let G be a group and let x ∈ G. The conjugate of x by g ∈ G is the element g −1 xg. Let X=G. Then G acts on X by conjugation, i.e. xg = g −1 xg. The orbit xG of x ∈ G in this action is called the conjugacy class of x. The stabilizer Gx = {g ∈ G : g −1 xg = x} is the set of elements that commute with x, and is called the centralizer of x in G. It is usually denoted by CG (x). If G|X is a group action and g ∈ G, we define Fix(g) by Fix(g)={x ∈ X|xg = x}. We denote |Fix(g)| by χG|X (g), or simply χ(g) if the group action is understood.
7 It is elementary to establish the following results (see [15, 16, 17, 18, 35]): Theorem 1.4 |xG | = [G : Gx ] = |G|/|Gx|. Theorem 1.5 The number of conjugates of x in G is [G : CG (g)] = |G|/|CG (x)|. Theorem 1.6 Gxg =(Gx )g . Theorem 1.7 Fix(g h)=(Fix(g))h, and therefore χ(g h )=χ(g). The following elementary result is due to Cauchy and Frobenius. It is commonly but erroneously referred to as Burnside’s lemma. Theorem 1.8 (Cauchy-Frobenius) Let a group G act on a finite set X. If N is the number of orbits of G, then N=
1 X χ(g). |G| g∈G
Let k be a positive integer. Then the action G|X induces an action G| obvious way: If x = {x1 , x2 , . . . , xk } ∈
X k
X k
in the
and g ∈ G, then xg = {xg1 , xg2 , . . . , xgk }.
Similarly, G acts on the k-tuples of X in a natural way: If ~x = (x1 , x2 , . . . , xk ) is a k-tuple of X, then ~xg = (xg1 , xg2 , . . . , xgk ). The group action G|X is said to be t-homogeneous if the action of G on the tsubsets of X is transitive. That is, G|X is t-homogeneous if for any pair of t-element subsets S1 , S2 of X, there is an element g ∈ G such that S1g = S2 .
8 The group G is said to be t-transitive on X if the induced action of G on the t-tuples of X is transitive. That is, if for any pair of t-tuples ~x, ~y of X, there is an element g ∈ G such that ~xg = ~y . A group is said to be of type t∗ if it is t-homogeneous, but not t-transitive. For simplicity, we call these t∗ -transitive. The subgroup G of SX is an automorphism group of a design (X, B) if for all g ∈ G and S ∈ B, S g ∈ B. In this case, we say the t-design is G-invariant. If S ⊆ X, the orbit of S is S G = {S g : g ∈ G} and the stabilizer of S is GS = {g ∈ G : S g = S}. It follows that G is an automorphism group of a design (X, B) if and only if B is a union of G-orbits on
X k
.
A system of distinct orbit representatives of the orbits that make up B is called a collection of base blocks. Example: It is evident from Figure 1 that the group G = h(034)(165), (15)(34)i is an automorphism group of the 2 − (7, 3, 1) design given in Section 1.2. Furthermore, under the group G, {013, 124, 156} are base blocks. Let t < k, and let G act on X. Let ∆ = {∆i }ri=1 be the G-orbits on Γ = {Γj }rj=1 be the G-orbits on
X k
X t
, and let
. For each ∆i , fix an arbitrary orbit representative
Ti . Let A∆,Γ = (ai,j ) be the matrix in which entry ai,j is the number of elements of
9 Γj which contain Ti . Although perhaps not immediately obvious, it is true that any representative from ∆i will be contained in the same number of elements of Γj . The following theorem is due to Kramer and Mesner.
Theorem 1.9 There exists a G-invariant t-(v,k,λ) if and only if there is a {0, 1}solution x to the matrix equation
A∆,Γ · x = λ(1, 1, . . . , 1)⊤ .
In other words, to find a t-design, one must find a set of columns of A∆,Γ whose row-sums are all equal to λ. Notice that if G is t-homogeneous, then there is only one orbit on t-subsets of X. Thus the matrix A∆,Γ has only one row. It follows that if G is a t-homogeneous group, then for any k ≥ t, any orbit of k-element subsets of G forms a t-(v,k,λ) design. Moreover, any union of such orbits is also a t-design. We denote by GF(q) the finite field of order q, where q is a prime power. The non-zero entries of a finite field form a multiplicative cyclic group GF(q)∗ of order q − 1. A primitive element (or primitive root) α ∈ GF(q) is an element that generates GF(q)∗ . For more information on group theory, the reader is directed to [35]. For more on automorphism groups of a design, see [5, 7].
10
1.4
Large Sets
The collection
X k
of all k-subsets of a set X forms a t-(v,k,λ) design with λ=λ=
v−t k−t
.
This is usually called the complete design. An LS[N](t, k, v), or a large set of t − (v, k, λ) designs, is a partition, [(X, Bi )]N i=1 , of the complete design into N disjoint t − (v, k, λ/N) designs. Large sets are also denoted LSλ (t, k, v) to place the emphasis on the value of λ. The λ is omitted if it is one. Recall that λ∗ is the minimal value of λ for which there exists a t−(v, k, λ) design. A large set of t − (v, k, λ∗ ) designs is called a (t, k, v)-decomposition. It should be noted that the original definition of a large set was restricted to only those for which λ = λ∗ . Let B = {Bi }N i=1 be the collection of designs in a large set. A group G is said to be an automorphism group of a large set if Bg =B for all g ∈ G, that is, if Big ∈ B for all Bi ∈ B and g ∈ G. Equivalently, we say that a large set with this property is G-invariant. If the stronger condition that Big =Bi for all Bi ∈ B and g ∈ G holds, we say that a large set is [G]-invariant. Let G be t-homogeneous on X with a semiregular induced action on
X k
. Then
the matrix A∆,Γ has only one row, and all entries are the same. Since each orbit is a
11 t-(v,k,λ) design, it is clear that there is an LS[N](t,k,v) with N=
v k
/|G|. In this case
we say the large set is G-semiregular. It is not too hard to see that a G-semiregular large set is [G]-invariant, and that a [G]-invariant large set is G-invariant. Theorems 1.10, 1.11, 1.12, and 1.13 give necessary and sufficient conditions for the existence of all three classes of large sets. Theorem 1.10 Let G be t-homogeneous on X. Then there exists a G-semiregular large set if and only if no element of prime order fixes any k-subsets of X. Proof. Let G be t-homogeneous on X. Assume that no element of prime order fixes any k-subsets of X. Then, if B ∈
X k
then GB =1 otherwise, if 1 6= g ∈ GB , there is
a power of g which is an element of prime order fixing B. Hence, |B G | =|G|, and the collection of all G-orbits of k-subsets is a G-semiregular large set. Conversely, if B={Bi }ri=1 is a semiregular large set of t − (v, k, λ) designs, then each Bi is a G−orbit of size |G|, and therefore if B ∈ Bi , we have GB = 1, so no elements of prime order fix B.
The following two theorems allude to two different techniques which can be used to find [G]-invariant large sets. The first is a simple extension of Theorem 1.9. Essentially it says that a [G]-invariant large set exists if and only if one can find N disjoint solutions to the matrix equations given in Theorem 1.9.
12 Theorem 1.11 A [G]-invariant LS[N]-(t,k,v) exists if and only if there is a {0, 1}solution Y with constant row sum 1 to the 1 .. A∆,Γ · Y = λ . 1
matrix equation 1 ... 1 .. .. . . 1 ... 1
|∆|,N
Theorem 1.12 essentially states that if all t-designs with a G as an automorphism group are known, then one needs simply to find a collection of these that partitions the orbits on k-subsets to find a [G]-invariant large set. The vector x selects the disjoint t-designs. Theorem 1.12 Let Y be a matrix whose rows are all solutions to the matrix equation given in Theorem 1.9. Then a [G]-invariant LS[N]-(t,k,v) exists if and only if there is a {0, 1}-solution x to the matrix equation xY = (1, 1, . . . , 1) Denote the orbits of
X k
under the action of G by Γ = {Γi : 1 ≤ i ≤ s}. Also, let
Θ = {Θi}ri=1 be the G-orbits on the collection of all t − (v, k, λ) designs with point set X. For each Θi , fix an arbitrary orbit representative Di . Define the r × s matrix M = mij by the rule mij = |Di ∩ Γj | × |Θi|/|Γj |. It should be noted that the entry mij is independent of the orbit representative Di that is chosen. Theorem 1.13 gives the necessary and sufficient conditions for the existence of G-invariant large sets.
13 Theorem 1.13 There exists a G-invariant large set of t − (v, k, λ) designs if and only if there exists a {0, 1}-solution x to the matrix equation xM = (1, 1, . . . , 1)⊤ .
Tables of the existence of t-designs and large sets for v ≤ 30 are given in [9], and in [27], the existence of large sets is completely determined for v ≤ 12. A survey on large sets is given by Teirlinck in [16]. In [11], the authors list all known parameter sets for large sets with v ≤ 18. The table indicates whether a large set with the given parameters exists, doesn’t exist, or is unknown.
Chapter 2 Transitive Groups 2.1
Examples of transitive groups
Let q = pa be a prime power, let X = GF (q), let V = V (n, q) be a vector space of dimension n over GF(q), and let N be an arbitrary set. Also, In is defined to be the n × n identity matrix. When the size is understood, it will be denoted simply by I. In the remainder of this section, we define some of the groups used in the remainder of the thesis. The definitions of the following groups, and more information about them, can be found in many sources, including Rotman’s text [35].
• SN is the symmetric group on the set N.
• AN , called the alternating group, is the subgroup of SN consisting of all even permutations.
14
15 • GL(n,q), the general linear group, is the multiplicative group of all nonsingular m × m matrices over GF (q).
• ZGL (n, q), the center of GL(n,q), is the subgroup consisting of all non-zero scalar transformations.
• SL(n,q), the special linear group, is the multiplicative group of all n×n matrices over GF (q) with determinant 1.
• ZSL (n, q), the center of SL(n,q), is the group of all n × n matrices of the form kI, with k n = 1.
• ΓL(n, q) is the group of all nonsingular semilinear transformations of V to itself under composition. The center of ΓL(n, q) is ZGL (n, q). • PGL(n,q) = GL(n, q)/ZGL(n, q), the projective linear group. • PSL(n,q) = SL(n, q)/ZSL (n, q), the projective special linear group, also called the projective unimodular group.
• PΓL(n,q) = ΓL(n, q)/ZGL (n, q), the collineation group. The following definitions can be found in [21].
16 • AΓL(1,q) is the group of mappings x 7→ axσ + b on GF(q), where a 6= 0 and b are in GF(q) and σ ∈ Aut(GF(q)).
• AGL(1,q), the affine group, is the group of mappings x 7→ ax + b on GF(q), where a 6= 0, and b are in GF(q).
Carter [8] gives descriptions of the following groups, including many references.
• PSU(3,q) is the projective special unitary group.
• PSp(2d, 2) is the projective symplectic group. This is a subgroup of PSL(2d, q).
• 2 B2 (q) is the Suzuki group. • 2 G2 (q) is the Ree group.
• HS is the Higman-Sims group.
• Co3 is the third Conway group.
• M11 , M12 , M22 , M23 , and M24 are the Mathieu groups.
2.2
A list of all t- and t∗-transitive groups
It is clear that any t-transitive group is t-homogeneous. The fact that for t ≥ 5 the converse is true was proved by Livingstone and Wagner [31]. Thus for t ≥ 5, there are
17 no t∗ -transitive groups. Also, if G is a t-transitive group of degree n, then G is also (n − t)-homogeneous. Thus, when speaking of t∗ -transitive groups, we will assume that n ≥ 2t. The following theorem due to Kantor [21] determines all t∗ -transitive groups.
Theorem 2.1 Let G be a group of type t∗ on a finite set of n points, where n ≥ 2t. Then up to permutation isomorphism, one of the following holds:
(i) t=2 and G ≤ AΓL(1, q) with n = q ≡ 3 mod 4;
(ii) t=3 and PSL(2, q) ≤ G ≤ P ΓL(2, q), where n − 1 = q ≡ 3 mod 4;
(iii) t=3 and G = AGL(1, 8), AΓL(1, 8) or AΓL(1, 32); or
(iv) t=4 and G = P SL(2, 8), P ΓL(2, 8) or P ΓL(2, 32).
All t- and t∗ -transitive groups are essentially known. Lists of 2-transitive groups are widely available [4, 22, 6]. Cameron [6] gives a list that includes enough information to classify all t- and t∗ -transitive groups. More precisely, all t- and t∗ -transitive groups are known for t > 3. 3- or 3∗ -transitive groups must have PSL(2,q) as a minimal normal subgroup, with a few exceptions. For t = 2, there is a longer list of minimal normal subgroups. Table 2.2 gives the characterization of t and t∗ -transitive
18 groups. Note that a group listed as t-transitive is not (necessarily) also listed as (t − 1)-transitive.
2.3
PSL(2,q)
The action of PSL(2,q)on the projective line X = GF(q) ∪ {∞} is well known (when q ≡ 3 mod 4 see [13], for example). The results below will be of use.
Theorem 2.2 PSL(2,q)is 3-homogeneous on X when q ≡ 3 mod 4.
Theorem 2.3 Let g ∈ PSL(2,q)be an element of order d. If g has an a-cycle, then a = d or a = 1.
Table 2.1 contains information about the elements of PSL(2,q). In the table, φ(n) is the number of numbers mod n relatively prime to n. 1
p
2
d| q−1 2
d| q+1 , d 6= 2 2
order of the centralizer of g
(q 3 −q) 2
q
q+1
(q−1) 2
(q+1) 2
number of conjugacy classes
1
2
1
φ(d)/2
φ(d)/2
q2 − 1
(q 2 −q)
φ(d)(q 2 +q)
2
2
φ(d)(q 2 −q) 2
1
0
2
0
order of g
number of elements
1
number of fixed points χ(g) q + 1
Table 2.1: Useful facts about PSL(2,q) when q ≡ 3 mod 4
19
If G is t- or t∗ -transitive on a set X, then: t Group degree t ≥ 6 G = Sk is k transitive k G = Ak is (k − 2)-transitive k A6 has two representations t∗ , t ≥ 5 There are none 5 G = M24 24 G = M12 (Two representations) 12 4 G = M23 23 G = M11 11 ∗ 4 G=PSL(2,8) 9 G=PΓL(2,8) 9 G=PΓL(2,32) 33 3 G = M22 22 G = M11 12 G has PSL(2,q) as a minimal normal subgroup q+1 ∗ 3 G=AGL(1,8) 8 G=AΓL(1,8) 8 G=AΓL(1,32) 32 PSL(2,q) ≤ G ≤ PΓL(2,q), q ≡ 3 mod 4 q+1 2 G has one of the following as a minimal normal subgroup (q d −1) PSL(d,q), d ≥ 2 (d,q) 6= (2,2), (2,3) (q−1) (Two representations when d > 2) PSU(3,q),q > 2 q3 + 1 2 2a+1 B2 (q) (Suzuki), q = 2 >2 q2 + 1 2 G2 (q) (Ree) , q = 32a+1 > 3 q3 + 1 2d−1 PSp(2d,2), d > 2 2 ± 2d−1 PSL(2,8) 28 PSL(2,11) (Two representations) 11 A7 (Two representations) 15 HS (Higman-Sims) (Two representations) 176 Co3 (Conway) 276 A = Zp × Zp × · · · × Zp , elementary abelian pn 2∗ G ≤ AΓL(1, q), q ≡ 3 mod 4 q Table 2.2: Finite t-transitive and t∗ -transitive groups
Chapter 3 The Existence of G-semiregular Large Sets The following sections give some examples of the existence and non-existence of Gsemiregular large sets. Section 3.1 settles the existence of [PSL(2, q)]-invariant 3designs, and Section 3.2 gives some examples of the non-existence of [G]-invariant large sets for the Mathieu Groups and a few linear groups.
3.1
PSL(2,q)
Theorem 3.1 At least one k-subset of X is fixed by at least one non-identity element g ∈ PSL(2,q) if and only if one of the following is true:
1. k is even,
2. q ≡ −1 mod d, for some prime d|k,
3. q ≡ 0 mod d, for some prime d|k(k − 1), or 20
21 4. q ≡ 1 mod d, for some prime d|k(k − 1)(k − 2), d 6= 2.
Proof.
Let g ∈ PSL(2,q) have prime order d > 1 and fix some k-subset of X. By
Table 2.1, there are four cases to consider. In each case, Table 2.1 gives the number of fixed points. Case 1: If d = 2, then g fixes no points of X, and consists only of 2-cycles. Thus g fixes some k-subset of X if and only if k is even. Case 2: If d| q+1 and d 6= 2, then g fixes no points of X and consists of only d-cycles. 2 In this case, some k-subset of X is fixed by g if and only if d|k. Also, notice that the condition d| q+1 is equivalent to q ≡ −1 mod d, since d 6= 2 is prime. Thus, g fixes 2 some k-subset of X if and only if q ≡ −1 mod d for some prime d|k. , then g consists of d-cycles and 2 fixed points on X. In this case, Case 3: If d| q−1 2 some k-subset of X is fixed by g if and only if d|k (if g fixes no points of the k-subset), d|(k − 1) (if g fixes one point of the k-subset), or d|(k − 2) (if g fixes two points of the k-subset). Also, notice that the condition d| q−1 is equivalent to q ≡ 1 mod d, since 2 d 6= 2 is prime. Thus, g fixes some k-subset of X if and only if q ≡ 1 mod d for some prime d|k(k − 1)(k − 2). Case 4: If d = p, where q = pe , then g consists of p-cycles and one fixed point on X. In this case, some k-subset of X is fixed by g if and only if d|k (if g fixes no
22 points of the k-subset), or d|(k − 1) (if g fixes one point of the k-subset). Thus g fixes some k-subset of X if and only if p|k(k − 1). Notice that this is equivalent to saying q ≡ 0 mod d for some d|k(k − 1).
Theorem 3.2 Let q = pe ≡ 3 mod 4. Then there exists a [PSL(2,q)]-semiregular LS[N](3,k,q+1), for N =
(q+1 k ) (q+1)q(q−1)/2
, if and only if the following conditions hold
simultaneously.
1. k is odd
2. q 6≡ −1 mod d, for all prime d|k
3. q 6≡ 0 mod d, for all prime d|k(k − 1)
4. q 6≡ 1 mod d, for all prime d|k(k − 1)(k − 2), d 6= 2 (q+1 k ) Proof. Assume there is a [PSL(2,q)]-semiregular LS[N](3,k,q+1) for N= (q+1)q(q−1)/2 . Then by Theorem 1.10 this is equivalent to the fact that no element of PSL(2,q)fixes any k-subset of X. By Lemma 3.1, all of the above conditions must be met.
Theorem 3.3 Let q = pe ≡ 3 mod 4, and q ≥ 2k. Then there exists a PSL(2,q)semiregular LS[N](3,k,q+1) for N = tions hold.
(q+1 k ) (q+1)q(q−1)/2
if and only if the following condi-
23 =1 1. k(k − 1)(k − 2), q−1 2 2. (k(k − 1), p) = 1 3. k, q+1 =1 2 Proof. We will show that this theorem is equivalent to Theorem 3.2. Let d be prime. Then
q+1 ,k 2
=1 ⇔ d6| ⇔
q+1 for all d|k 2
q+1 6≡ 0 mod d for all d|k 2
⇔ q 6≡ −1 mod d for all d|k, and k is odd.
(q, k(k − 1)) = 1 ⇔ d 6 |q, for all d|k(k − 1) ⇔ q 6≡ 0 mod d for all d|k(k − 1).
q−1 , k(k − 1)(k − 2) 2
=1 ⇔ d6| ⇔
q−1 for all d|k(k − 1)(k − 2) 2
q−1 6≡ 0 mod d for all d|k(k − 1)(k − 2) 2
⇔ q 6≡ 1 mod d for all d|k(k − 1)(k − 2), d 6= 2, since q 6≡ 1 mod 2.
24 Lemma 3.1 Let q = pe ≡ 3 mod 4 satisfy the sufficient conditions of 3.2. Then q = 3e or q ≡ 11 mod 12.
Proof.
When q ≡ 3 mod 4, then q ≡ 3, 7 or 11 mod 12. If q ≡ 3 mod 12, then
q=3 + 12s=3(1 + 4s). Since q is a prime power, q = 3e is evident. If q ≡ 7 mod 12, then q ≡ 1 mod 3, but condition 3) from Theorem 3.2 requires q 6≡ 1 mod 3, since clearly 3|k(k − 1)(k − 2). Thus, q 6≡ 7 mod 12. Thus, either q ≡ 11 mod 12, or q = 3e . Lemma 3.2 If q ≡ 11 mod 12 satisfies the sufficient conditions of Theorem 3.2, then k ≡ 1 or 5 mod 6.
Proof.
Notice that condition 1) from Theorem 3.2 requires that q 6≡ 1 mod 2, and
q 6≡ 2 mod 3. But q ≡ 11 mod 12 implies q ≡ 1 mod 2 and q ≡ 2 mod 3.
Note, that since q ≡ 11 mod 12 implies q ≡ 1 mod 2 and q ≡ 2 mod 3, and since k ≡ 1 or 5 mod 6, the primes in Theorem 3.2 can be assumed to be greater than 3.
Theorem 3.4 Let k ≡ 1 or 5 mod 6. Then there are infinitely many primes q which satisfy the sufficient conditions of Theorem 3.2.
Proof. Let q ≡ 11 mod 12. Then the primes 2 and 3 don’t restrict k in Theorem 3.2, Q so can be ignored. Let k(k − 1)(k − 2)=( nr=1 ri ei ) 2s 3t , where the ri > 3 are distinct
25 primes, and set r = r1 r2 · · · rn . Then the equations q 6≡ 0, ±1 mod d, for all d|k(k − 1)(k −2) imply the conditions of Theorem 3.2, so produce large sets. By Corollary 1.1 there are (r1 − 3)(r2 − 3) · · · (rn − 3) solutions modulo r to these equations. Since the ri > 3, this product is non-zero. By Dirichlet’s Theorem, each of these solutions produces infinitely many solutions.
Although Theorem 3.4 gives infinitely many large sets, one might ask if this is a very sparse family. It is easy to write an algorithm to test the conditions of Theorem 3.2 for all congruence classes modulo r. Table 3.1 gives a short list of the number of solutions modulo r, and the density, defined as the number of solutions divided by 4 × φ(r), since the primes fall uniformly into the φ(r) residue classes modulo r, and since we also require that p ≡ 11 mod 12, which constitutes a quarter of the primes. Table A.1 in Appendix A gives a larger table, including k ≤ 223. A few comments are in order. First, notice that for most k in the table, the density is usually somewhere between .10 and .20, which is a significant percentage of the primes. When considering the fact that only one quarter of the primes (those p for which p ≡ 11 mod 12) even have a chance of satisfying the necessary conditions, the density seems even more impressive. Another thing to notice is that this doesn’t include prime powers, for which there are certainly many more large sets. In order
k k(k-1)(k-2) r φ(r) Θ(3.2) Θ(3.4) Density 5 60 5 4 2 2 0.125000 7 210 35 24 12 8 0.125000 11 990 55 40 24 16 0.150000 13 1716 143 120 90 80 0.187500 17 4080 85 64 42 28 0.164063 19 5814 323 288 240 224 0.208333 23 10626 1771 1320 900 640 0.170455 25 13800 115 88 42 40 0.119318 29 21924 203 168 130 104 0.193452 31 26970 4495 3360 2268 1456 0.168750 35 39270 6545 3840 1080 896 0.070313 37 46620 1295 864 510 272 0.147569 41 63960 2665 1920 1254 760 0.163281 43 74046 12341 10080 7800 6080 0.193452 47 97290 5405 4048 2772 1760 0.171196 49 110544 329 276 180 176 0.163044
26
Table 3.1: The density of p for which there exists a PSL(2,p)-semiregular large sets.
to get an estimate of the density of large sets from prime powers, one would need to know the density of prime powers modulo n for a given n, which, to my knowledge, is unknown. When q = 3e , I haven’t shown as strong a result, but a few things of interest can be said. Lemma 3.3 If q = 3e satisfies the sufficient conditions of Theorem 3.2, then k ≡ 5 mod 6. Proof. The sufficient conditions of Theorem 3.2 require that (k(k − 1), 3e ) = 1 and ) = 1 The first equation implies k ≡ 2 mod 3, and the second that k ≡ 1 mod 2 (k, q+1 2
27 since
q+1 2
≡ 0 mod 2. Together, these give k ≡ 5 mod 6.
Theorem 3.5 Let q = 3e , e odd. Then a set of sufficient conditions for the existence of a PSL(2,q)-semiregular LS[N](3,k,q+1), with N =
(q+1 k ) (q+1)q(q−1)/2
, is
1. k(k − 1)(k − 2), q−1 =1 2 =1 2. k, q+1 2 3. k ≡ 5 mod 6
Proof.
Theorem 3.3 shows that the sufficient conditions from Theorem 3.2 imply
those of this theorem. Now, when q = 3e , and k ≡ 5 mod 6, it is clear that (k(k − 1), 3e ) = 1. Thus the conditions above are sufficient.
3.2
t- and t∗-transitive groups with t ≥ 3
When discussing transitive groups in this section, the groups Sn and An are omitted since they are not interesting in the present context. We also concern ourselves only with those groups which are at least 3∗ -transitive. In this section we show that their are no semiregular large sets for t ≥ 4, except perhaps from PΓL(2, 32), with 10 ≤ k ≤ 16, and the only semiregular large sets for t = 3 come from groups with PSL(2,q) as a subgroup, and possibly AΓL(2, 32).
28 If G is a group which is at least 4∗ -transitive, then G is one of M24 , M23 , M12 , M11 , PSL(2,8), PΓL(2, 8), or PΓL(2, 32). If G is 3- or 3∗ -transitive, then G has PSL(2, q) as a subgroup, or G is one of M22 , M11 (acting on 12 points), AGL(1,8), AΓL(1, 8), or AΓL(1, 32). It is not too difficult to determine the existence or non-existence of semiregular large sets from the groups M24 , M23 , M12 , M11 , M22 , M11 (acting on 12 points), PSL(2,8), PΓL(2, 8), PΓL(2, 32), AGL(1,8), AΓL(1, 8), and AΓL(1, 32). One can compute the orbits on k sets and easily determine whether or not the G-orbits have length |G|. In fact, one can also compute the A∆,Γ matrix for the permissible k and determine whether there exist [G]-invariant large sets. The Mathieu Groups yield no semiregular large sets. For M22 , M23 , and M24 , this is clear from [25]. It is easy to compute A∆,Γ for M11 and M12 in their usual representations, and M11 acting 12 points, and come to the same conclusion. Notice that PSL(2, 8) and PΓL(2,8) are 4∗ -transitive, and of degree 9, so they are set-transitive. In other words, there is only 1 orbit of k-subsets for 1 ≤ k ≤ 9, and there are no semiregular large sets from these groups. Computing the A∆,Γ matrices for AGL(1, 8), AΓL(1,8) is very simple, and shows easily that there are no semiregular large sets for these groups.
29 I have verified for 5 ≤ k ≤ 9 that there are no PΓL(2, 32)-semiregular large sets, and I suspect this will also hold for 10 ≤ k ≤ 16 as well. I have not done any computations for AΓL(1,32), although it should not be too difficult. From Table 2.2 and the above discussion, it is clear that their are no semiregular large sets for t ≥ 4, except perhaps from PΓL(2, 32), with 10 ≤ k ≤ 16, and the only semiregular large sets for t = 3 come from groups with PSL(2,q) as a subgroup, and possibly AΓL(2, 32).
Chapter 4 Recursive Constructions for Large Sets 4.1
Known Recursive Constructions
There are several known recursive constructions for large sets. Two are obvious. The first comes from the fact that the existence of a LS[N](t, k, v) implies the existence of a LS[M](t, k, v) for all M dividing N. The second comes from the fact that the existence of an LS[N](t, k, v) implies the existence of a LS[N](t − 1, k − 1, v − 1) by taking the derived design with respect to a particular fixed point x0 of each design in the large set. Several other recursive constructions for large sets of t-designs are known. The following list is taken from [11].
Theorem 4.1 If there exist an LS[M](t, k, v) and an LS[N](t, k + 1, v), then there exists an LS[gcd(M, N)](t, k + 1, v + 1).
30
31 Theorem 4.2 (Teirlinck [41]) For every natural number t let λ(t) = lcm{ lcm{1, 2, . . . , t + 1}, and let ℓ(t) =
Qt
i=1
t m
: m = 1, 2, . . . , t}, let λ∗ (t) =
λ(i) · λ∗ (i). Then, for all N > 0, there is an
LS[N](t, t + 1, t + N · ℓ(t)).
Theorem 4.3 (Khosrovshahi and Ajoodani-Namini [23]) If there are LS[N](t, t + 1, v) and LS[N](t, t + 1, w), then there is also an LS[N](t, t + 1, v + w − t).
Theorem 4.4 (Qiu-rong Wu [42]) If there exist large sets LS[N](t, k, v), LS[N](t, k, w), LS[N](k − 2, k − 1, v − 1), LS[N](k − 2, k − 1, w − 1), then there exists a large set LS[N](t, k, v + w − k + 1).
Corollary 4.1 If there exist large sets LS[N](t, k, v) and LS[N](k − 2, k − 1, v − 1), then there exist large sets LS[N](t, k, v + m(v − k + 1)), for all m ≥ 0.
Theorem 4.5 (Ajoodani-Namini [3] ) If there exists an LS[N](t, m, v − 1), and mN < k < (m + 1)N, then a LS[N](t + 1, k, Nv) also exists.
32
4.2
Applying the Recursive Constructions
Applying the recursive constructions is very elementary, since it only involves looking at the parameters of a large set, not at the actual structures themselves. I have written a program that applies the recursive constructions. Starting with all known large sets having v ≤ 18, of which there are approximately 100, the above constructions yield about 300 new large sets with v ≤ 50. When v is allowed to be as large as 300, these constructions yield about 5600 new large sets. Table 4.1 gives the list of the known large sets with v ≤ 18, plus all large sets with v ≤ 30 which are constructed from these base sets. Table 4.1: Large sets with v ≤ 30 parameters LS[2](2,3,6) LS[2](2,3,10) LS[2](2,3,14) LS[2](2,3,18) LS[2](2,3,22) LS[2](2,3,26) LS[2](2,3,30) LS[2](2,4,10) LS[2](2,4,11) LS[2](2,4,18) LS[2](2,4,19) LS[2](2,4,27) LS[2](2,5,10) LS[2](2,5,11) LS[2](2,5,12) LS[2](2,5,18) LS[2](2,5,19) LS[2](2,5,20) LS[2](2,5,28) LS[2](2,6,12)
λ 2 4 6 8 10 12 14 14 18 60 68 150 28 42 60 280 340 408 1300 105
Method Operands Bhattacharya LS [4](2,3,10) exists 2—4 Hanani LS [8](2,3,18) exists 2—8 KAN LS[2](2,3,6) LS[2](2,3,18) KAN LS[2](2,3,10) LS[2](2,3,18) KAN LS[2](2,3,14) LS[2](2,3,18) LS [14](2,4,10) exists 2—14 LS [6](2,4,11) exists 2—6 LS [10](2,4,18) exists 2—10 Chee LS[2](2,3,18) LS[2](2,4,18) Wu LS[2](2,4,11) LS[2](2,4,19) LS[2](2,3,10) LS[2](2,3,18) LS [14](2,5,10) exists 2—14 LS [42](2,5,11) exists 2—42 LS [6](2,5,12) exists 2—6 Magliveras and Laue Chee LS[2](2,4,18) LS[2](2,5,18) Wu LS[2](2,5,12) LS[2](2,5,12) LS[2](3,4,11) LS[2](3,4,11) Wu LS[2](2,5,12) LS[2](2,5,20) LS[2](3,4,11) LS[2](3,4,19) LS [42](2,6,12) exists 2—42
parameters LS[2](2,6,13) LS[2](2,6,18) LS[2](2,6,19) LS[2](2,6,20) LS[2](2,6,21) LS[2](2,6,29) LS[2](2,7,14) LS[2](2,7,18) LS[2](2,7,19) LS[2](2,7,20) LS[2](2,7,21) LS[2](2,7,22) LS[2](2,7,30) LS[2](2,8,21) LS[2](2,8,22) LS[2](2,8,23) LS[2](2,9,18) LS[2](2,9,20) LS[2](2,9,22) LS[2](2,9,23) LS[2](2,9,24) LS[2](2,10,23) LS[2](2,10,24) LS[2](2,10,25) LS[2](2,11,24) LS[2](2,11,25) LS[2](2,11,26) LS[2](3,4,11) LS[2](3,4,19) LS[2](3,4,27) LS[2](3,5,11) LS[2](3,5,12) LS[2](3,5,20) LS[2](3,5,28) LS[2](3,6,12) LS[2](3,6,13) LS[2](3,6,21) LS[2](3,6,29) LS[2](3,7,14) LS[2](3,7,22) LS[2](3,7,30) LS[2](3,8,21) LS[2](3,9,22) LS[2](3,10,21) LS[2](3,10,23) LS[2](3,11,24)
λ 165 910 1190 1530 1938 8775 396 2184 3094 4284 5814 7752 49140 13566 19380 27132 5720 15912 38760 58140 85272 101745 159885 245157 248710 408595 653752 4 8 12 14 18 68 150 42 60 408 1300 165 1938 8775 4284 13566 15912 38760 101745
Method Operands LS [6](2,6,13) exists 2—6 Magliveras and Laue Chee LS[2](2,5,18) LS[2](2,6,18) Chee LS[2](2,5,19) LS[2](2,6,19) Wu LS[2](2,6,13) LS[2](2,6,13) LS[2](4,5,12) Wu LS[2](2,6,13) LS[2](2,6,21) LS[2](4,5,12) LS [12](2,7,14) exists 2—12 LS [8](2,7,18) exists 2—8 Chee LS[2](2,6,18) LS[2](2,7,18) Chee LS[2](2,6,19) LS[2](2,7,19) Chee LS[2](2,6,20) LS[2](2,7,20) Wu LS[2](2,7,14) LS[2](2,7,14) LS[2](5,6,13) Wu LS[2](2,7,14) LS[2](2,7,22) LS[2](5,6,13) DER LS[2](3,9,22) Chee LS[2](2,7,21) LS[2](2,8,21) Chee LS[2](2,7,22) LS[2](2,8,22) LS [10](2,9,18) exists 2—10 DER LS[2](3,10,21) DER LS[2](3,10,23) Chee LS[2](2,8,22) LS[2](2,9,22) Chee LS[2](2,8,23) LS[2](2,9,23) DER LS[2](3,11,24) Chee LS[2](2,9,23) LS[2](2,10,23) Chee LS[2](2,9,24) LS[2](2,10,24) DER LS[2](3,12,25) Chee LS[2](2,10,24) LS[2](2,11,24) Chee LS[2](2,10,25) LS[2](2,11,25) derivation of LS[4](4,5,12 KAN LS[2](3,4,11) LS[2](3,4,11) Wu LS[2](3,4,11) LS[2](3,4,19) LS[2](2,3,10) Magliveras and Laue LS [6](3,5,12) exists 2—6 Wu LS[2](3,5,12) LS[2](3,5,12) LS[2](3,4,11) Wu LS[2](3,5,12) LS[2](3,5,20) LS[2](3,4,11) LS [42](3,6,12) exists 2—42 LS [6](3,6,13) exists 2—6 Wu LS[2](3,6,13) LS[2](3,6,13) LS[2](4,5,12) Wu LS[2](3,6,13) LS[2](3,6,21) LS[2](4,5,12) Magliveras and Laue Wu LS[2](3,7,14) LS[2](3,7,14) LS[2](5,6,13) Wu LS[2](3,7,14) LS[2](3,7,22) LS[2](5,6,13) DER LS[2](4,9,22) DER LS[2](4,10,23) DER LS[2](4,11,22) DER LS[2](4,11,24) DER LS[2](4,12,25)
33
LS[2](4,5,12) LS[2](4,5,20)
LS[2](5,6,13) LS[2](5,6,21)
LS[2](2,3,18)
LS[2](3,4,11) LS[2](3,4,19)
LS[2](4,5,12) LS[2](4,5,20) LS[2](5,6,13) LS[2](5,6,21)
parameters LS[2](3,12,25) LS[2](4,5,12) LS[2](4,5,20) LS[2](4,5,28) LS[2](4,6,12) LS[2](4,6,13) LS[2](4,6,21) LS[2](4,6,29) LS[2](4,7,14) LS[2](4,7,22) LS[2](4,7,30) LS[2](4,9,22) LS[2](4,10,23) LS[2](4,11,22) LS[2](4,11,24) LS[2](4,12,25) LS[2](4,13,26) LS[2](5,6,13) LS[2](5,6,21) LS[2](5,6,29) LS[2](5,7,14) LS[2](5,7,22) LS[2](5,7,30) LS[2](5,11,24) LS[2](5,12,25) LS[2](5,13,26) LS[2](6,7,14) LS[2](6,7,22) LS[2](6,7,30) LS[2](6,13,26) LS[3](2,3,11) LS[3](2,3,20) LS[3](2,3,29) LS[3](2,4,11) LS[3](2,4,12) LS[3](2,4,21) LS[3](2,4,30) LS[3](2,5,11) LS[3](2,5,12) LS[3](2,5,13) LS[3](2,5,22) LS[3](2,6,12) LS[3](2,6,13) LS[3](2,6,14) LS[3](2,6,23) LS[3](2,7,14)
λ 248710 4 8 12 14 18 68 150 60 408 1300 4284 13566 15912 38760 101745 248710 4 8 12 18 68 150 13566 38760 101745 4 8 12 38760 3 6 9 12 15 57 126 28 40 55 380 70 110 165 1995 264
Method Operands DER LS[2](4,13,26) Denniston KAN LS[2](4,5,12) LS[2](4,5,12) Wu LS[2](4,5,12) LS[2](4,5,20) Magliveras and Laue Magliveras and Laue Wu LS[2](4,6,13) LS[2](4,6,13) Wu LS[2](4,6,13) LS[2](4,6,21) Magliveras and Laue Wu LS[2](4,7,14) LS[2](4,7,14) Wu LS[2](4,7,14) LS[2](4,7,22) AN LS[2](3,4,11) DER LS[2](5,11,24) AN LS[2](3,5,11) AN LS[2](3,5,12) DER LS[2](5,13,26) AN LS[2](3,6,13) derivation of LS[4](6,7,14 KAN LS[2](5,6,13) LS[2](5,6,13) Wu LS[2](5,6,13) LS[2](5,6,21) Magliveras and Laue Wu LS[2](5,7,14) LS[2](5,7,14) Wu LS[2](5,7,14) LS[2](5,7,22) AN LS[2](4,5,12) DER LS[2](6,13,26) AN LS[2](4,6,13) Kreher and Radziszowski KAN LS[2](6,7,14) LS[2](6,7,14) Wu LS[2](6,7,14) LS[2](6,7,22) AN LS[2](5,6,13) Teirlinck KAN LS[3](2,3,11) LS[3](2,3,11) DER LS[3](3,4,30) LS [6](2,4,11) exists 3—6 LS [15](2,4,12) exists 3—15 Wu LS[3](2,4,12) LS[3](2,4,12) Wu LS[3](2,4,12) LS[3](2,4,21) LS [42](2,5,11) exists 3—42 LS [6](2,5,12) exists 3—6 Magliveras and Laue Wu LS[3](2,5,13) LS[3](2,5,13) LS [42](2,6,12) exists 3—42 LS [6](2,6,13) exists 3—6 Magliveras and Laue Wu LS[3](2,6,14) LS[3](2,6,14) LS [12](2,7,14) exists 3—12
34
LS[2](3,4,11) LS[2](3,4,19)
LS[2](4,5,12) LS[2](4,5,12) LS[2](4,5,12) LS[2](4,5,20) LS[2](5,6,13) LS[2](5,6,13) LS[2](5,6,13) LS[2](5,6,21)
LS[2](4,5,12) LS[2](4,5,20) LS[2](5,6,13) LS[2](5,6,13) LS[2](5,6,13) LS[2](5,6,21)
LS[2](5,6,13) LS[2](5,6,21)
LS[3](2,3,11) LS[3](2,3,11) LS[3](2,3,11) LS[3](2,3,20)
LS[3](3,4,12) LS[3](3,4,12)
LS[3](4,5,13) LS[3](4,5,13)
parameters LS[3](2,7,15) LS[3](3,4,12) LS[3](3,4,21) LS[3](3,4,30) LS[3](3,5,12) LS[3](3,5,13) LS[3](3,5,22) LS[3](3,6,12) LS[3](3,6,13) LS[3](3,6,14) LS[3](3,6,23) LS[3](3,7,14) LS[3](3,7,15) LS[3](4,5,13) LS[3](4,5,22) LS[3](4,6,13) LS[3](4,6,14) LS[3](4,6,23) LS[3](4,7,14) LS[3](4,7,15) LS[4](2,3,10) LS[4](2,3,18) LS[4](2,3,26) LS[4](2,7,14) LS[4](2,7,18) LS[5](2,3,12) LS[5](2,3,22) LS[5](2,4,8) LS[5](2,4,12) LS[5](2,4,13) LS[5](2,4,18) LS[5](2,4,23) LS[5](2,9,18) LS[5](3,4,13) LS[5](3,4,23) LS[6](2,4,11) LS[6](2,5,11) LS[6](2,5,12) LS[6](2,6,12) LS[6](2,6,13) LS[6](2,7,14) LS[6](3,5,12) LS[6](3,6,12) LS[6](3,6,13) LS[7](2,3,9) LS[7](2,3,16)
λ 429 3 6 9 12 15 57 28 40 55 380 110 165 3 6 12 15 57 40 55 2 4 6 198 1092 2 4 3 9 11 24 42 2288 2 4 6 14 20 35 55 132 6 14 20 1 2
Method Operands Chee LS[3](2,6,14) LS[3](2,7,14) Teirlinck KAN LS[3](3,4,12) LS[3](3,4,12) Wu LS[3](3,4,12) LS[3](3,4,21) LS [6](3,5,12) exists 3—6 Chee, Colbourn, Furino, Kreher Wu LS[3](3,5,13) LS[3](3,5,13) LS [42](3,6,12) exists 3—42 LS [6](3,6,13) exists 3—6 Magliveras and Laue Wu LS[3](3,6,14) LS[3](3,6,14) Magliveras and Laue Chee LS[3](3,6,14) LS[3](3,7,14) Kramer, Magliveras and O’Brien KAN LS[3](4,5,13) LS[3](4,5,13) Magliveras and Laue Chee, Colbourn, Furino, Kreher Wu LS[3](4,6,14) LS[3](4,6,14) Magliveras and Laue Chee LS[3](4,6,14) LS[3](4,7,14) Teirlinck LS [8](2,3,18) exists 4—8 KAN LS[4](2,3,10) LS[4](2,3,18) LS [12](2,7,18) exists 4—12 LS [8](2,7,18) exists 4—8 Schreiber KAN LS[5](2,3,12) LS[5](2,3,12) Sharry and Street LS [15](2,4,12) exists 5—15 LS [55](2,4,13) exists 5—55 LS [10](2,4,18) exists 5—10 Wu LS[5](2,4,13) LS[5](2,4,13) LS [10](2,9,18) exists 5—10 Kramer, Magliveras and O’Brien KAN LS[5](3,4,13) LS[5](3,4,13) Chee, Colbourn, Furino, Kreher LS [42](2,5,11) exists 6—42 LS[6](3,5,12 as 2- designs LS [42](2,6,12) exists 6—42 Chee, Magliveras LS [12](2,7,14) exists 6—12 Kramer, Magliveras and Stinson LS [42](3,6,12) exists 6—42 Chee, Magliveras Kirkman Magliveras and Laue
35
LS[3](2,3,11) LS[3](2,3,20)
LS[3](3,4,12) LS[3](3,4,12)
LS[3](4,5,13) LS[3](4,5,13)
LS[3](4,5,13) LS[3](4,5,13)
LS[5](2,3,12) LS[5](2,3,12)
parameters LS[7](2,3,23) LS[7](2,3,30) LS[7](2,4,9) LS[7](2,4,10) LS[7](2,4,17) LS[7](2,4,24) LS[7](2,5,10) LS[7](2,5,11) LS[7](2,5,18) LS[7](2,6,12) LS[7](3,4,17) LS[7](3,5,10) LS[7](3,6,12) LS[11](2,3,13) LS[11](2,3,24) LS[11](2,4,13) LS[11](2,4,14) LS[11](2,4,25) LS[11](2,5,13) LS[11](2,5,14) LS[11](2,5,15) LS[11](2,6,13) LS[11](2,6,14) LS[11](2,6,15) LS[11](2,6,16) LS[13](2,3,15) LS[13](2,3,28) LS[14](2,4,10) LS[14](2,5,10) LS[14](2,5,11) LS[14](2,6,12) LS[14](3,6,12) LS[15](2,4,12) LS[21](2,5,11) LS[21](2,6,12) LS[21](3,6,12) LS[42](2,5,11) LS[42](2,6,12) LS[42](3,6,12) LS[55](2,4,13)
λ 3 4 3 4 15 33 8 12 80 30 2 3 12 1 2 5 6 23 15 20 26 30 45 65 91 1 2 2 4 6 15 6 3 4 10 4 2 5 2 1
Method Operands KAN LS[7](2,3,9) LS[7](2,3,16) KAN LS[7](2,3,16) LS[7](2,3,16) Kramer, Magliveras and Stinson LS [14](2,4,10) exists 7—14 Wu LS[7](2,4,10) LS[7](2,4,10) LS[7](2,3,9) LS[7](2,3,9) Wu LS[7](2,4,10) LS[7](2,4,17) LS[7](2,3,9) LS[7](2,3,16) LS [14](2,5,10) exists 7—14 LS [42](2,5,11) exists 7—42 Magliveras and Laue LS [14](2,6,12) exists 7—42 Chee, Magliveras extension of LS[7](2,4,9 LS [42](3,6,12) exists 7—42 Denniston KAN LS[11](2,3,13) LS[11](2,3,13) LS [55](2,4,13) exists 11—55 Kramer, Magliveras and O’Brien Wu LS[11](2,4,14) LS[11](2,4,14) LS[11](2,3,13) LS[11](2,3,13) Chee, Magliveras Chee, Magliveras Chee, Magliveras Chee, Magliveras Chee LS[11](2,5,13) LS[11](2,6,13) Chee LS[11](2,5,14) LS[11](2,6,14) Chee LS[11](2,5,15) LS[11](2,6,15) Denniston KAN LS[13](2,3,15) LS[13](2,3,15) Kramer, Magliveras and Stinson Kramer, Magliveras and Stinson LS [42](2,5,11) exists 14—42 LS [42](2,6,12) exists 14—42 DIV LS[42](3,6,12) Kramer, Magliveras and Stinson LS [42](2,5,11) exists 21—42 LS [42](2,6,12) exists 21—42 LS [42](3,6,12) exists 21—42 Kramer, Magliveras and Stinson LS [42](3,6,12 as 2- designs extension of LS [42](2,5,11 Chouinard
36
Chapter 5 Further Research There are many areas for further research in large sets, particularly related to several of the topics of this thesis. In terms of PSL(2,q) being an automorphism group of a large set, what more can be said? We have settled the question for PSL(2,q)semiregular large sets, but there may be other [PSL(2, q)]-invariant or PSL(2, q)invariant large sets. For example, the existence of [PSL(2, q)]-invariant large sets for k = 4 and k = 5 has been settled [14]. It should be noted that the [PSL(2, q)]-invariant large sets for k = 5 are actually PSL(2,q)-semiregular. One might ask whether there is a general result for [PSL(2, q)]-invariant large sets with arbitrary k. In finding G-semiregular large sets, one only considers those matrices A∆,Γ which have one row. It has been suggested by others that it may be worth while to consider the case when A∆,Γ has two or three rows. As shown in Table 2.2, there are many t- and t∗ - transitive groups which may yield 37
38 G-semiregular or [G]-invariant large sets. For t ≥ 3, the only groups which have not been considered extensively are PΓL(2,32), AΓL(1,32), and groups with PSL(2, q) as a subgroup. It seems unlikely that there are PΓL(2, 32)-semiregular large sets, although there may be [P ΓL(2, 32)]-invariant large sets. I suspect the same is true of AΓL(1,32). With a little work, the groups PΓL(2,32) and AΓL(1,32) can be settled, leaving the groups with PSL(2, q) as a subgroup to consider. There are also many 2and 2∗ -transitive groups to consider. Magliveras and Laue, among others, have proposed a large database containing the known large sets. The database I have created is very modest in terms of capability and size. To be complete, one should add to the database some of the other known large sets, including several infinite families. In addition, other recursive constructions for large sets should be investigated. In particular, none of the known constructions increases the parameter N. Thus, most of the constructions increase λ, making the chance of finding a large set with minimal λ using a construction small. Although I am skeptical that such a construction exists, it certainly is worth considering.
Appendix A Density of Large Sets from PSL(2,q) Table A.1 gives the density of primes q for which there exist PSL(2,q)-semiregular large sets, for k ≤ 223. r is the product of prime divisors of k(k − 1)(k − 2) greater than 3, Θ(3.2) and Θ(3.4) are the number of solutions modulo r due to Theorems 3.2 and 3.4, respectively, and Density is Θ(3.2)/(4 × φ(r)).
39
40 Table A.1: The density of p for which there exists a PSL(2,p)-semiregular large sets. k k(k-1)(k-2) r φ(r) Θ(3.2) Θ(3.4) Density 5 60 5 4 2 2 0.125000 7 210 35 24 12 8 0.125000 11 990 55 40 24 16 0.150000 13 1716 143 120 90 80 0.187500 17 4080 85 64 42 28 0.164063 19 5814 323 288 240 224 0.208333 23 10626 1771 1320 900 640 0.170455 25 13800 115 88 42 40 0.119318 29 21924 203 168 130 104 0.193452 31 26970 4495 3360 2268 1456 0.168750 35 39270 6545 3840 1080 896 0.070313 37 46620 1295 864 510 272 0.147569 41 63960 2665 1920 1254 760 0.163281 43 74046 12341 10080 7800 6080 0.193452 47 97290 5405 4048 2772 1760 0.171196 49 110544 329 276 180 176 0.163044 53 140556 11713 9984 8250 7000 0.206581 55 157410 2915 2080 816 800 0.098077 59 195054 32509 29232 25704 23296 0.219828 61 215940 17995 13920 9918 6496 0.178125 65 262080 455 288 100 80 0.086805 67 287430 47905 31680 19008 10240 0.150000 71 342930 57155 36960 21420 10880 0.144886 73 373176 5183 5040 4830 4760 0.239583 77 438900 7315 4320 1632 1024 0.094445 79 474474 79079 56160 37620 24320 0.167468 83 551286 3403 3280 3120 3040 0.237805 85 592620 49385 31488 11340 8960 0.090034 89 681384 28391 24640 20898 17888 0.212033 91 728910 40495 25344 10440 6880 0.102983 95 830490 138415 99360 41760 39424 0.105072 97 884640 9215 6912 4794 3008 0.173394 101 999900 5555 4000 2646 1568 0.165375 103 1061106 176851 163200 148500 137200 0.227482 107 1190910 198485 132288 79560 41600 0.150354 109 1259604 11663 11448 11130 11024 0.243056
k k(k-1)(k-2) r φ(r) Θ(3.2) Θ(3.4) 113 1404816 29267 24192 19250 14960 115 1481430 246905 177408 75480 70400 119 1642914 91273 66816 35112 31360 121 1727880 6545 3840 1800 896 125 1906500 6355 4800 2262 2128 127 2000250 4445 3024 1860 992 131 2196870 366145 262080 173184 102400 133 2299836 191653 140400 74304 65536 137 2515320 11645 8704 6030 3752 139 2627934 437989 412896 385560 364480 143 2863146 477191 386400 248400 239360 145 2985840 20735 13440 5148 4160 149 3241644 38591 31968 25550 19856 151 3374850 112495 88800 65268 43216 155 3652110 202895 115200 37800 25088 157 3796260 316355 224640 147378 86240 161 4095840 42665 27456 12240 8000 163 4251366 26243 21384 16800 12800 167 4574130 762355 544480 358668 209920 169 4741464 15197 11952 8250 6560 173 5088276 141341 130032 118490 108800 175 5267850 175595 115584 36936 35360 179 5639574 939929 908512 872784 847616 181 5831820 161995 128160 94518 62656 185 6229320 259555 190080 84252 78880 187 6434670 1072445 691200 341040 213248 191 6858810 127015 82080 47940 24064 193 7077696 36863 36480 35910 35720 197 7529340 89635 56448 32010 15520 199 7762194 431233 388080 343980 304192 203 8242206 1373701 1108800 669240 652288 205 8489460 707455 430080 153900 110656 209 8998704 62491 47520 29568 25600 211 9260790 1543465 907200 477360 212992 215 9800130 1633355 1246560 579600 565760 217 10077480 46655 30240 13776 8960 221 10647780 887315 552960 268380 156800 223 10940826 1823471 1534464 1270500 1047200
Density 0.198930 0.106365 0.131376 0.117188 0.117813 0.153770 0.165202 0.132308 0.173196 0.233449 0.160714 0.095759 0.199809 0.183750 0.082031 0.164016 0.111451 0.196409 0.164684 0.172565 0.227809 0.079890 0.240169 0.184375 0.110811 0.123351 0.146016 0.246094 0.141768 0.221591 0.150893 0.089460 0.155556 0.131548 0.116240 0.113889 0.121338 0.206994
41
42 k
k(k-1)(k-2) r φ(r) Θ(3.2) Θ(3.4) Density
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