Locally primitive Cayley graphs of finite simple ... - Semantic Scholar
VoL 44 No. I
SCIENCE IN CHINA (Series A)
January 2001
Locally primitive Cayley graphs of finite simple groups FANG Xingui ( ~ , ~ - ) 1 ,
C. E. Praeger 2 & WANG Jie (_.T.. ,~,,)l
l. Department of Mathematics, Peking University, Beijing 100871, China 2. Department of Mathematics, The University of Western Australia Perth, WA 6907, Australia Correspondence should be addressed to Fang Xingui Received June 3, 2000 Abstract A graph /- is said to be G-locally primitive, where G is a subgroup of automorphisms of /-, if the stabiliser G~ of a vertex a acts primitively on the s e t / - ( a ) of vertices of/- adjacent to or. For a finite non-abelian simple group L and a Cayiey subset S of L, suppose that L IOut(L)l +1. ( b ) r ~< ( r e ( L ) - 2 ) / 2 , for any odd prime divisor r of I O u t ( L ) I . ( c ) If l K I = I L l , then K = L . Proof. Part ( a ) holds immediately from refs. [ 6 , 7 ] (see also [ 8 ] ) . For part ( b ) , suppose that I O u t ( L ) I is divisible by an odd prime r . Then L is a simple group of Lie type over a field of order q = pa for some prime p and integer a I> 1. The result follows immediately from part ( a ) unless l O u t ( L ) I = r . Hence we assume that I O u t ( L ) l = r . If a = 1, it is easy to show that ( b ) holds. If a > 1, we may assume I O u t ( L ) I = r = a , because a divides I O u t ( L ) l . Direct checking of these remaining cases shows that for all such L , we have r ~< ( m ( L ) - 2 ) / 2 . Now assume that I K 1 = I L I with K ~< Aut( L ) . Since L 4, (PI2s+ ( q ) , P / 2 7 ( q ) ) . Now I V/-"I = I T : R Iz. I R 2 : W~ I and Wa is a subdirect product of R x R . Since also I V/" I = I L I = I T I , we find that in none of these possibilities for ( T, R ) is there a suitable subgroup W~. This contradiction completes the proof. Finally we prove that A u t ( / ' ) has socle L. This proposition, together with Propositions 1 and 2, completes the proof of Theorem 1. Proposition 3. Suppose that G and P are as in Theorem 1. Then Aut/" is almost simple with socle L. P r o o f . Suppose that the socle of Aut(/-') is not L = soc( G ) . Without loss of generality we may assume that G = A u t ( L ) n A u t ( / - ' ) . LetY be a subgroup of Aut(/-') such that G is maximal in Y. By Proposition 2, Y is almost simple with socle M , say, and M contains L. From the choice of G we know that L # M . Therefore M~ # ! and, since F' is connected, Mv~(~ is a nontrivial normal subgroup of the primitive group Y~. So M~ ~> is transitive. Let B be a maximal subgroup of Y containing Y~. If B contained M , then we would have Y = MY,~ ~ B , which is
No. 1
LOCALLY PRIMITIVE CAYLEY GRAPHS OF FINITE SIMPLE GROUPS
65
not true. Thus Y = GB is a factorisation of Y such that both G and B are maximal subgroups of Y, and neither G nor B contains M = s o c ( Y ) . Suppose first that /-' has valency 4. Then L = PI2 s+ ( q ) with q odd, and Y~ is a t 2 , 3 1 group. Hence I M: LI = 2a3 b for some a ~ > 0 , b I>0. If M is not an alternating group, then a similar checking process to that used in the proof of Lemma 7 shows that there is no subgroup Y in tables 1 - - 6 of ref. [13] with a maximal subgroup G having socle Pl-2s+ ( q ) ,
and with I M:
LI = 2 a 3 b. Thus M = A~ for some n~>5, and by Theorem D o f r e f . [ 1 3 ] , since L = Pl'28+ ( q ) (which has no 2-transitive representation), the subgroup B is A~_ l or S~_ 1 and G has a transitive representation of degree n . Thus n I> m ( L ) t> ( q3 + 1 ) ( q4 _ 1 ) ( s e e , for example, ref. [17]).
By Bertrand's postulate Els] , there is a prime l such that n / 2 < l r dividing IM : L I, which is a contradiction. Suppose that M = Am, for some n i> 5. Since Y = GB with G an almost simple group of Lie type with socle L , (and with I G : L I divisible by the odd prime r ) , it follows from Theorem D o f r e f . [13] that either ( i ) n = 1 0 , L = L 2 ( 8 ) (so r = 3 ) a n d A s x A s < ~ B < ~ S s w r S2with B transitive of degree 10, or (ii) A,_ k ~< B ~< Sn_ k x Sk for some k with 1 ~< k m ( L ) , so r ~ < ( n - 2 ) / 2 by Lemma 3 ( b ) . Also, since I O u t ( L ) I has an odd prime divisor, it follows that m ( L ) I>9, so n i>9. By Bertrand's postulate EIs~ , there is a prime l such that ( n - 1 ) / 2 < l~< ( n - 1) - 2 . Such a prime l divides I M I , and is greater than r , and hence does not divide I Y: L I . Thus l divides I L I . If l > n / 2 , then the transitive group G is primitive of degree n , and since l < n 3 it follows from refs. [ 1 9 , 1 3 . 1 0 ] that G contains A n , which is not the case. Hence l = n~ 21>5, and for n > 1 1 0 , there is a second prime l' such that n / 2 < l ' ~ < n 3, and we o b t a i n a contradiction as before. -
Thus ( M , L ) = ( U 4 ( 3 ) , L 3 ( 4 ) ) ,
so I M : L I
= 2 . 3 4 and I O u t ( M ) l
=8.
Thus l e v i =
I Y : L I divides 24 . 34 and is divisible by 2.34 . Since r divides I Y~ I we must have r = 3. Thus Y~
~< $3, and it follows from ref. [20] that I Y, I = 3 . 2 ~ for some a . This contradicts the fact
that 34 divides I Y~I . [] Finally in this section we give an example of several G- locally primitive Cayley graphs it' for the simple group L = s o c ( G ) = S z ( 8 ) for which G = A u t ( / - ' ) , to demonstrate that such graphs exist. E x a m p l e 1. Let G = Sz ( 8 ) . 3 and take x to be an involution of Sz ( 8 ) and y to be the field automorphism of order 3 such t h a t <x,~fY,xY"> .~ S z , ( 8 ) . Let /-' = C a y ( S z ( 8 ) , S ) with S 2
= I x , x y , x y t . Then a small computation shows that the Cayley graph /-' = C a y ( S z ( 8 ) , S ) has full automorphism group A u t ( F ' ) = S z ( 8 ) . 3 and that there are three non-isomorphic graphs corresponding to different choices of the involution x . Their diameters and girths are as follows:
66
SCIENCE IN CHINA (Series A)
Vol. 44
Diameter
17
18
21
Girth
15
12
10
Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant No. 69873002).
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
Fang, X. G . , Praeger, C. E . , On graphs admitting arc-transitive actions of almost simple groups, J. Algebra, 1998, 205 (1): 37. Fang, X. G . , Havas, G . , Praeger, C. E . , On the automorphism groups of quasiprimitive almost simple graphs, J. Algebra, 1999, 222(2): 271. Biggs, N . , Algebraic graph theory, 2nd e d . , London: Cambridge University Press, 1993. Godsil, C. D. , On the full automorphism group of a graph, Combinatoriea, 1981, 1(2) : 243. Praeger, C. E . , Imprimitive symmetric graphs, Ars Combinatoria, 1985, 19A(1): 149. Liebeek, M. W. , On the orders of maximal subgroups of the finite classical groups, Proc. London Math. Soe. ( 3 ) , 1985, 50(2) : 426. Liebeek, M. W . , Saxl, J . , On the orders of maximal subgroups of the finite exceptional groups of Lie type, J. London Math. Soc. ( 2 ) , 1987, 55(2) : 299. Cooperstein, B. N . , Minimal degree for a permutation representation of a classical group, Israel J. Math., 1978, 30(2) : 213. Conway, J. H . , Curtis, R. T . , Norton, S. P. et al., Atlas of finite groups, Oxford: Clarendon Press, 1985. Gorenstein, D . , Lyons, R . , The local structure of finite groups of characteristic 2 type, Memoirs Amer. Math soc., 1983, 42(276) : 1. Zsigmondy, K . , Zur Theorie der Potenzreste, Monatsh Math. Phys., 1892, 3(2) : 265. Huppert, B . , Blackburn, N . , Finite Groups II, Berlin: Springer-Verlag, 1982. Liebeck, M. W . , Praeger, C. E . , Saxl, J . , The maximal factorisations of the almost simple groups and their automorphism groups, Memoirs Amer. Math. Soe., 1990, 86(432): 1. Praeger, C. E . , On the O' Nan-Scott Theorem for finite quasiprimitive permutation groups and an application to 2-arc transitive graphs, J. London Math. Soc. ( 2 ) , 1993, 47(1): 227. Praeger, C. E . , Finite transitive permutation groups and finite vertex-transitive graphs, Graph Symmetry: algebraic methods and applications. (Hahn, G . , Sabidussi, G . ) , Dordrecht: Kluwer, 1997, 277--318. Kimmerle, W . , Lyons, R . , Sandling. R. et al., Composition factors from the group ring and Artin' s theorem on orders of simple groups, Proc. London Math. Soc. ( 3 ) , 1990, 60(1): 89. Kleidman, P. B . , The maximal subgroups of the finite 8-dimensional orthogonal groups Pf2s~ ( q ) and of their automorphism groups, J. Algebra, 1987, 110(1): 173. Hall, M. J r . , The Theory of Groups, New York: MacMiUan, 1959. Wielandt, H . , Finite Permutation Groups, New York: Academic Press, 1964. Weiss, R . , An application of p-factorization methods to symmetric graphs, Math. Proc. Camb. Phil; Soc., 1979, 85(1) : 43.
SCIENCE IN CHINA (Series A)
January 2001
Locally primitive Cayley graphs of finite simple groups FANG Xingui ( ~ , ~ - ) 1 ,
C. E. Praeger 2 & WANG Jie (_.T.. ,~,,)l
l. Department of Mathematics, Peking University, Beijing 100871, China 2. Department of Mathematics, The University of Western Australia Perth, WA 6907, Australia Correspondence should be addressed to Fang Xingui Received June 3, 2000 Abstract A graph /- is said to be G-locally primitive, where G is a subgroup of automorphisms of /-, if the stabiliser G~ of a vertex a acts primitively on the s e t / - ( a ) of vertices of/- adjacent to or. For a finite non-abelian simple group L and a Cayiey subset S of L, suppose that L IOut(L)l +1. ( b ) r ~< ( r e ( L ) - 2 ) / 2 , for any odd prime divisor r of I O u t ( L ) I . ( c ) If l K I = I L l , then K = L . Proof. Part ( a ) holds immediately from refs. [ 6 , 7 ] (see also [ 8 ] ) . For part ( b ) , suppose that I O u t ( L ) I is divisible by an odd prime r . Then L is a simple group of Lie type over a field of order q = pa for some prime p and integer a I> 1. The result follows immediately from part ( a ) unless l O u t ( L ) I = r . Hence we assume that I O u t ( L ) l = r . If a = 1, it is easy to show that ( b ) holds. If a > 1, we may assume I O u t ( L ) I = r = a , because a divides I O u t ( L ) l . Direct checking of these remaining cases shows that for all such L , we have r ~< ( m ( L ) - 2 ) / 2 . Now assume that I K 1 = I L I with K ~< Aut( L ) . Since L 4, (PI2s+ ( q ) , P / 2 7 ( q ) ) . Now I V/-"I = I T : R Iz. I R 2 : W~ I and Wa is a subdirect product of R x R . Since also I V/" I = I L I = I T I , we find that in none of these possibilities for ( T, R ) is there a suitable subgroup W~. This contradiction completes the proof. Finally we prove that A u t ( / ' ) has socle L. This proposition, together with Propositions 1 and 2, completes the proof of Theorem 1. Proposition 3. Suppose that G and P are as in Theorem 1. Then Aut/" is almost simple with socle L. P r o o f . Suppose that the socle of Aut(/-') is not L = soc( G ) . Without loss of generality we may assume that G = A u t ( L ) n A u t ( / - ' ) . LetY be a subgroup of Aut(/-') such that G is maximal in Y. By Proposition 2, Y is almost simple with socle M , say, and M contains L. From the choice of G we know that L # M . Therefore M~ # ! and, since F' is connected, Mv~(~ is a nontrivial normal subgroup of the primitive group Y~. So M~ ~> is transitive. Let B be a maximal subgroup of Y containing Y~. If B contained M , then we would have Y = MY,~ ~ B , which is
No. 1
LOCALLY PRIMITIVE CAYLEY GRAPHS OF FINITE SIMPLE GROUPS
65
not true. Thus Y = GB is a factorisation of Y such that both G and B are maximal subgroups of Y, and neither G nor B contains M = s o c ( Y ) . Suppose first that /-' has valency 4. Then L = PI2 s+ ( q ) with q odd, and Y~ is a t 2 , 3 1 group. Hence I M: LI = 2a3 b for some a ~ > 0 , b I>0. If M is not an alternating group, then a similar checking process to that used in the proof of Lemma 7 shows that there is no subgroup Y in tables 1 - - 6 of ref. [13] with a maximal subgroup G having socle Pl-2s+ ( q ) ,
and with I M:
LI = 2 a 3 b. Thus M = A~ for some n~>5, and by Theorem D o f r e f . [ 1 3 ] , since L = Pl'28+ ( q ) (which has no 2-transitive representation), the subgroup B is A~_ l or S~_ 1 and G has a transitive representation of degree n . Thus n I> m ( L ) t> ( q3 + 1 ) ( q4 _ 1 ) ( s e e , for example, ref. [17]).
By Bertrand's postulate Els] , there is a prime l such that n / 2 < l r dividing IM : L I, which is a contradiction. Suppose that M = Am, for some n i> 5. Since Y = GB with G an almost simple group of Lie type with socle L , (and with I G : L I divisible by the odd prime r ) , it follows from Theorem D o f r e f . [13] that either ( i ) n = 1 0 , L = L 2 ( 8 ) (so r = 3 ) a n d A s x A s < ~ B < ~ S s w r S2with B transitive of degree 10, or (ii) A,_ k ~< B ~< Sn_ k x Sk for some k with 1 ~< k m ( L ) , so r ~ < ( n - 2 ) / 2 by Lemma 3 ( b ) . Also, since I O u t ( L ) I has an odd prime divisor, it follows that m ( L ) I>9, so n i>9. By Bertrand's postulate EIs~ , there is a prime l such that ( n - 1 ) / 2 < l~< ( n - 1) - 2 . Such a prime l divides I M I , and is greater than r , and hence does not divide I Y: L I . Thus l divides I L I . If l > n / 2 , then the transitive group G is primitive of degree n , and since l < n 3 it follows from refs. [ 1 9 , 1 3 . 1 0 ] that G contains A n , which is not the case. Hence l = n~ 21>5, and for n > 1 1 0 , there is a second prime l' such that n / 2 < l ' ~ < n 3, and we o b t a i n a contradiction as before. -
Thus ( M , L ) = ( U 4 ( 3 ) , L 3 ( 4 ) ) ,
so I M : L I
= 2 . 3 4 and I O u t ( M ) l
=8.
Thus l e v i =
I Y : L I divides 24 . 34 and is divisible by 2.34 . Since r divides I Y~ I we must have r = 3. Thus Y~
~< $3, and it follows from ref. [20] that I Y, I = 3 . 2 ~ for some a . This contradicts the fact
that 34 divides I Y~I . [] Finally in this section we give an example of several G- locally primitive Cayley graphs it' for the simple group L = s o c ( G ) = S z ( 8 ) for which G = A u t ( / - ' ) , to demonstrate that such graphs exist. E x a m p l e 1. Let G = Sz ( 8 ) . 3 and take x to be an involution of Sz ( 8 ) and y to be the field automorphism of order 3 such t h a t <x,~fY,xY"> .~ S z , ( 8 ) . Let /-' = C a y ( S z ( 8 ) , S ) with S 2
= I x , x y , x y t . Then a small computation shows that the Cayley graph /-' = C a y ( S z ( 8 ) , S ) has full automorphism group A u t ( F ' ) = S z ( 8 ) . 3 and that there are three non-isomorphic graphs corresponding to different choices of the involution x . Their diameters and girths are as follows:
66
SCIENCE IN CHINA (Series A)
Vol. 44
Diameter
17
18
21
Girth
15
12
10
Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant No. 69873002).
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
Fang, X. G . , Praeger, C. E . , On graphs admitting arc-transitive actions of almost simple groups, J. Algebra, 1998, 205 (1): 37. Fang, X. G . , Havas, G . , Praeger, C. E . , On the automorphism groups of quasiprimitive almost simple graphs, J. Algebra, 1999, 222(2): 271. Biggs, N . , Algebraic graph theory, 2nd e d . , London: Cambridge University Press, 1993. Godsil, C. D. , On the full automorphism group of a graph, Combinatoriea, 1981, 1(2) : 243. Praeger, C. E . , Imprimitive symmetric graphs, Ars Combinatoria, 1985, 19A(1): 149. Liebeek, M. W. , On the orders of maximal subgroups of the finite classical groups, Proc. London Math. Soe. ( 3 ) , 1985, 50(2) : 426. Liebeek, M. W . , Saxl, J . , On the orders of maximal subgroups of the finite exceptional groups of Lie type, J. London Math. Soc. ( 2 ) , 1987, 55(2) : 299. Cooperstein, B. N . , Minimal degree for a permutation representation of a classical group, Israel J. Math., 1978, 30(2) : 213. Conway, J. H . , Curtis, R. T . , Norton, S. P. et al., Atlas of finite groups, Oxford: Clarendon Press, 1985. Gorenstein, D . , Lyons, R . , The local structure of finite groups of characteristic 2 type, Memoirs Amer. Math soc., 1983, 42(276) : 1. Zsigmondy, K . , Zur Theorie der Potenzreste, Monatsh Math. Phys., 1892, 3(2) : 265. Huppert, B . , Blackburn, N . , Finite Groups II, Berlin: Springer-Verlag, 1982. Liebeck, M. W . , Praeger, C. E . , Saxl, J . , The maximal factorisations of the almost simple groups and their automorphism groups, Memoirs Amer. Math. Soe., 1990, 86(432): 1. Praeger, C. E . , On the O' Nan-Scott Theorem for finite quasiprimitive permutation groups and an application to 2-arc transitive graphs, J. London Math. Soc. ( 2 ) , 1993, 47(1): 227. Praeger, C. E . , Finite transitive permutation groups and finite vertex-transitive graphs, Graph Symmetry: algebraic methods and applications. (Hahn, G . , Sabidussi, G . ) , Dordrecht: Kluwer, 1997, 277--318. Kimmerle, W . , Lyons, R . , Sandling. R. et al., Composition factors from the group ring and Artin' s theorem on orders of simple groups, Proc. London Math. Soc. ( 3 ) , 1990, 60(1): 89. Kleidman, P. B . , The maximal subgroups of the finite 8-dimensional orthogonal groups Pf2s~ ( q ) and of their automorphism groups, J. Algebra, 1987, 110(1): 173. Hall, M. J r . , The Theory of Groups, New York: MacMiUan, 1959. Wielandt, H . , Finite Permutation Groups, New York: Academic Press, 1964. Weiss, R . , An application of p-factorization methods to symmetric graphs, Math. Proc. Camb. Phil; Soc., 1979, 85(1) : 43.
