Generating sets of finite groups
aa r X i v : . [ m a t h . G R ] S e p GENERATING SETS OF FINITE GROUPS
PETER J. CAMERON, ANDREA LUCCHINI, AND COLVA M. RONEY-DOUGAL
Abstract.
We investigate the extent to which the exchange relationholds in finite groups G . We define a new equivalence relation ≡ m ,where two elements are equivalent if each can be substituted for theother in any generating set for G . We then refine this to a new sequence ≡ ( r )m of equivalence relations by saying that x ≡ ( r )m y if each can besubstituted for the other in any r -element generating set. The relations ≡ ( r )m become finer as r increases, and we define a new group invariant ψ ( G ) to be the value of r at which they stabilise to ≡ m .Remarkably, we are able to prove that if G is soluble then ψ ( G ) ∈{ d ( G ) , d ( G ) + 1 } , where d ( G ) is the minimum number of generators of G , and to classify the finite soluble groups G for which ψ ( G ) = d ( G ).For insoluble G , we show that d ( G ) ≤ ψ ( G ) ≤ d ( G ) + 5. However, weknow of no examples of groups G for which ψ ( G ) > d ( G ) + 1.As an application, we look at the generating graph of G , whose ver-tices are the elements of G , the edges being the 2-element generatingsets. Our relation ≡ (2)m enables us to calculate Aut(Γ( G )) for all solublegroups G of nonzero spread, and give detailed structural informationabout Aut(Γ( G )) in the insoluble case. Introduction
It is well known that generating sets for groups are far more complicatedthan generating sets for, say, vector spaces. The latter satisfy the exchangeaxiom, and hence any two irredundant sets have the same cardinality. Ac-cording to the Burnside Basis Theorem, a similar property holds for groupsof prime power order.Our starting point is the observation that, in order to understand betterthe generating sets for arbitrary finite groups, we should investigate theextent to which the exchange property holds. We define an equivalencerelation ≡ m on a finite group G , in which two elements are equivalent ifeach can be substituted for the other in any generating set for G . Then twoelements are equivalent if and only if they lie in the same maximal subgroupsof G .We refine this relation to a sequence of relations ≡ ( r )m whose terms dependon a positive integer r , where two elements are equivalent if each can be Date : September 21, 2016.
Key words and phrases. finite group, generation, generating graph.The second and third authors were supported by Universit`a di Padova (Progetto diRicerca di Ateneo: Invariable generation of groups). substituted for the other in any r -element generating set. The relations ≡ ( r )m become finer as r increases; we observe in Lemma 2.4 that the smallest valueof r for which ≡ ( r )m is not the universal relation is the minimum number d ( G )of generators of G .We define a new group invariant ψ ( G ) to be the value of r at which therelations ≡ ( r )m stabilise to ≡ m . Remarkably, it turns out (see Corollary 2.12)that if G is soluble then ψ ( G ) ∈ { d ( G ) , d ( G ) + 1 } . In Theorem 2.21 we evensucceed in giving a precise structural description of the finite soluble groups G for which ψ ( G ) = d ( G ).In the general case, we show in Corollary 2.13 and Proposition 2.14 that ψ ( G ) ≤ d ( G ) + 5, with tighter bounds when G is (almost) simple. However,we know of no examples of groups G for which ψ ( G ) > d ( G ) + 1.The relation ≡ m can be a little tricky to work with, so in Section 3 weintroduce a far simpler relation, by defining x ≡ c y if h x i = h y i . This isclearly a refinement of ≡ m , and provides an easy-to-calculate upper boundon the number of ≡ m -classes, and lower bound on their sizes. In Theorem 3.4we characterise the soluble groups G on which these two relations coincide;it would be very interesting to determine for which insoluble groups theyare equal.As an application, we notice that the relation ≡ (2)m is particularly inter-esting for two-generator groups. Such groups G have long been studied bymeans of the generating graph , whose vertices are the elements of G , theedges being the 2-element generating sets. The generating graph was de-fined by Liebeck and Shalev in [16], and has been further investigated bymany authors: see for example [3, 5, 6, 12, 18, 19, 20, 23] for some of therange of questions that have been considered. Many deep structural resultsabout finite groups can be expressed in terms of the generating graph.We notice that two group elements are ≡ (2)m -equivalent if and only if theyhave the same neighbours in the generating graph. By identifying the ver-tices in each equivalence class, we obtain a reduced graph Γ( G ), which hasmany fewer vertices, but the same spread, clique number and chromaticnumber, amongst other properties. We conjecture that in a group G ofnonzero spread, the equivalence relations ≡ m and ≡ (2)m coincide.The automorphism groups of generating graphs are extremely large, andtheir study has up to now seemed intractable. However, we show in Theo-rem 5.2 that the automorphism group of Γ( G ) has a very compact descrip-tion in terms of the sizes of the ≡ (2)m -classes of G , and the group Aut(Γ( G )).Using this, we are able to give a precise description of the automorphismgroups of the generating graphs of all soluble groups of nonzero spread, anda detailed description in the insoluble case.We have carried out many computational experiments on small insolublegroups G of nonzero spread. In each case we found that ψ ( G ) = 2, and that ENERATING SETS OF FINITE GROUPS 3
Aut(Γ( G )) is completely and straightforwardly determined by the sizes ofthe ≡ (2)m -classes and Aut( G ).The paper is structured as follows. In Section 2 we study the relations ≡ m and ≡ ( r )m , and the related invariant ψ ( G ). In Section 3 we look at therelation ≡ c . In Section 4 we introduce the generating graph Γ( G ) and thereduced generating graph Γ( G ), and then in Section 5 we study the groupAut(Γ( G )) for groups G of nonzero spread.2. A hierarchy of equivalences
Definitions and elementary results.
We shall now introduce ourmain families of relations, and establish a few basic results concerning them.
Definition 2.1.
Let G be a finite group. We define an equivalence relation ≡ m (m for “maximal subgroups”) on G by letting x ≡ m y if and only if x and y lie in exactly the same maximal subgroups of G .Note that the ≡ m -class containing the identity is precisely the Frattinisubgroup of G , and any ≡ m -class is a union of cosets of the Frattini subgroup.The equivalence relation ≡ m can also be characterised by a substitutionproperty: Proposition 2.2.
Let G be a finite group, and let x and y be elements of G . Then x ≡ m y if and only if ( ∀ r )( ∀ z , . . . , z r ∈ G )(( h x, z , . . . , z r i = G ) ⇔ ( h y, z , . . . , z r i = G )) . Proof.
Suppose first that h x, z , . . . , z r i = G but h y, z , . . . , z r i 6 = G . Thenthere is a maximal subgroup M of G containing y, z , . . . , z r . Clearly x / ∈ M ;so x m y .Conversely, suppose that x m y , so that (without loss of generality)there is a maximal subgroup M containing y but not x . Choose genera-tors z , . . . , z r for M . Then h y, z , . . . , z r i = M , but h x, z , . . . , z r i properlycontains M , and so is equal to G . (cid:3) This means that, when considering generating sets (of any cardinality)for a group G , we may restrict our attention to subsets of a set of ≡ m -classrepresentatives. Definition 2.3.
For any positive integer r , define equivalence relations ≡ ( r )m by the rule that x ≡ ( r )m y if and only if( ∀ z , . . . , z r − ∈ G )(( h x, z , . . . , z r − i = G ) ⇔ ( h y, z , . . . , z r − i = G )) . Lemma 2.4. (1)
The relations ≡ ( r )m get finer as r increases. (2) The smallest value of r for which ≡ ( r )m is not the universal relationis d ( G ) . For r = d ( G ) , there are at least r + 1 equivalence classes. (3) The limit value of this sequence of relations is ≡ m . PETER J. CAMERON, ANDREA LUCCHINI, AND COLVA M. RONEY-DOUGAL
Proof. (1) Choosing z r − to be the identity we see that x ≡ ( r )m y implies x ≡ ( r − y .(2) The first claim is clear, for this second, notice that the identity andthe elements of any d ( G )-element generating set are pairwise inequivalent.(3) This is clear. (cid:3) Definition 2.5.
Let ψ ( G ) be the value of r for which the equivalences ≡ ( r )m stabilise, that is, the least r such that ≡ ( r )m coincides with the limitingrelation ≡ m .2.2. Bounds on ψ ( G ) . In this subsection, we prove various upper and lowerbounds on ψ ( G ) in terms of other numerical invariants of G . We start withsome straightforward lower bounds on ψ ( G ). Lemma 2.6.
Let G be a finite group, and let d = d ( G ) . Then ψ ( G ) ≥ d ,and if G has a normal subgroup N such that N Frat( G ) and d ( G/N ) = d ,then ψ ( G ) ≥ d + 1 .Proof. The first claim is immediate from Lemma 2.4(2). For the second,notice that elements of N lie in no d -element generating set of G , and so are ≡ ( d )m -equivalent to the identity. However, the ≡ m -equivalence class of theidentity is Frat( G ). (cid:3) These lower bounds are best possible in a very strong sense: we know ofno groups that do not attain them.
Problem 2.7.
Is it true that if G is a finite group, then ψ ( G ) ∈ { d ( G ) , d ( G )+1 } ?Whilst we are not able to answer this question in general, in the restof this subsection we prove some upper bounds on ψ ( G ). In particular, inCorollary 2.12 we show that if G is soluble then ψ ( G ) ≤ d ( G ) + 1. Definition 2.8.
Let G be a finite group and let M be a core-free maximalsubgroup of G . For every g ∈ G \ M , let δ G,M ( g ) be the smallest cardinalityof a subset X of M with the property that G = h g, X i and let ν M ( G ) = sup g / ∈ M δ G,M ( g ) . Notice that ν M ( G ) ≤ d ( M ). Definition 2.9.
Let ˜ m ( G ) be the maximum of ν M/N ( G/N ) over all maximalsubgroups M of G , where N = Core G ( M ). Theorem 2.10. ψ ( G ) ≤ max { ˜ m ( G ) , d ( G ) } + 1 . Before proving this result, we briefly recall a necessary definition andresult. Given a subset X of a finite group G, we will denote by d X ( G ) thesmallest cardinality of a set of elements of G generating G together with theelements of X. The following generalizes a result originally obtained by W.Gasch¨utz [10] for X = ∅ . ENERATING SETS OF FINITE GROUPS 5
Lemma 2.11 ([6] Lemma 6) . Let X be a subset of G and N a normalsubgroup of G and suppose that h g , . . . , g k , X i N = G. If k ≥ d X ( G ) , thenthere exist n , . . . , n k ∈ N so that h g n , . . . , g k n k , X i = G. Proof of Theorem 2.10.
Let t = max { ˜ m ( G ) , d ( G ) } . Since the relations ≡ ( r )m become finer with r , it suffices to prove that if x and y are two elements of G and x m y, then x ( t +1)m y. So assume that x m y. It is not restrictive toassume that there exists a maximal subgroup M of G such that x / ∈ M and y ∈ M. Let N = Core G ( M ) and let X = { x } . Since t ≥ ˜ m ( G ) , we have t ≥ ν M/N ( G/N ) , hence there exist g , . . . , g t ∈ M such that h x, g , . . . , g t i N = G. Moreover t ≥ d ( G ) ≥ d X ( G ) . So we deduce from Lemma 2.11 that thereexist n , . . . , n t ∈ N such that G = h x, g n , . . . , g t n t i . On the other hand h y, g n , . . . , g t n t i ≤ M. Hence x ( t +1)m y. (cid:3) We are now able to prove a tight upper bound on ψ ( G ) for all finite solublegroups G . Corollary 2.12. If G is a finite soluble group, then ψ ( G ) ≤ d ( G ) + 1 . Proof.
Let M be a maximal subgroup of G , and let K = Core G ( M ). Then˜ G = G/K is a soluble group with a faithful primitive action on the cosets of
M/K , and d ( ˜ G ) ≤ d ( G ) . Moreover
M/K is a complement in ˜ G of Soc( ˜ G ) , so ν M/K ( G/K ) ≤ d ( M/K ) = d ( ˜ G/ Soc( ˜ G )) ≤ d ( ˜ G ) ≤ d ( G ) . This holds forevery maximal subgroup of G , so ˜ m ( G ) ≤ d ( G ) and the conclusion followsfrom Theorem 2.10. (cid:3) Now we prove an upper bound on ψ ( G ) for an arbitrary finite group G . Corollary 2.13. If G is a finite group, then ψ ( G ) ≤ d ( G )+5 . Furthermore,if G is simple, then ψ ( G ) ≤ , and if G is almost simple then ψ ( G ) ≤ .Proof. Burness, Liebeck and Shalev prove (see [4, Theorem 7]) that thepoint stabiliser of a d -generated finite primitive permutation group can begenerated by d + 4 elements. Hence if G is a finite group, then ˜ m ( G ) ≤ d ( G ) + 4 and our first claim follows from Theorem 2.10.In the same paper (see [4, Theorems 1 and 2]) they show that any maximalsubgroup of a finite simple group can be generated by 4 elements, and thatany maximal subgroup of an almost simple group can be generated by 6elements. Hence our final two claims follow in the same way. (cid:3) We conclude this subsection by mentioning a relationship with anotherwell-known parameter, µ ( G ), the maximum size of a minimal generating setfor G (a generating set for which no proper subset generates), studied byDiaconis and Saloff-Coste, Whiston, Saxl, and others [9, 14, 27]. Proposition 2.14.
Let G be a finite group. Then ψ ( G ) ≤ µ ( G ) . Hence if G = PSL ( p ) with p
6∈ { , , , } then ψ ( G ) ≤ , and ψ (PSL ( p )) ≤ in the remaining cases. PETER J. CAMERON, ANDREA LUCCHINI, AND COLVA M. RONEY-DOUGAL
Proof.
To prove that ψ ( G ) ≤ µ ( G ), we show that if µ = µ ( G ), and x ≡ ( µ )m y ,then x ≡ m y . So suppose that x ≡ ( µ )m y , and let G = h x, z , . . . , z r − i . Case r ≤ µ . Since the relations ≡ ( r )m get finer as r increases, in this case G = h y, z , . . . , z r − i . Case r > µ . In this case, our generating set is larger than µ , and sosome element is redundant. If x is redundant, then G = h z , . . . , z r − i = h y, z , . . . , z r − i , as required. Suppose that x is not redundant. Then G isgenerated by a subset of the given generators of size µ including x , withoutloss of generality { x, z , . . . , z µ − } . Since, by assumption, x ≡ ( µ )m y , we have G = h y, z , . . . , z µ − i = h y, z , . . . , z r − i .The final claim follows from [14], where the stated bounds on µ (PSL ( p ))are determined. (cid:3) In general µ ( G ) can be much larger than d ( G ). For example, if G issoluble, than m ( G ) − d ( G ) ≥ π ( G ) − µ ( G ) is at least the number of complemented factors in a chief series of G (see [17, Theorem 1]). Hence the difference µ ( G ) − d ( G ) (and consequently,by Corollary 1.10, the difference µ ( G ) − ψ ( G )) can be arbitrarily large.2.3. Groups with ψ ( G ) = d ( G ) . In this subsection, we study groups G forwhich ψ ( G ) = d ( G ); in particular in Theorem 2.21 we describe the structureof such soluble groups G . Definition 2.15.
A finite group G is efficiently generated if for all x ∈ G , d { x } ( G ) = d ( G ) implies that x ∈ Frat( G ) . Lemma 2.16. If ψ ( G ) = d ( G ) , then G is efficiently generated.Proof. Let d = d ( G ) . If G is not efficiently generated, then there exists x / ∈ Frat( G ) such that d { x } ( G ) = d. This implies in particular x ≡ ( d )m . However since x / ∈ Frat( G ) , we have x m
1, hence ψ ( G ) > d. (cid:3) Lemma 2.17. If G is efficiently generated and ˜ m ( G ) < d ( G ) , then ψ ( G ) = d ( G ) . Proof.
Let d = d ( G ). By Theorem 2.10, our assumption that ˜ m ( G ) < d ( G )implies that ψ ( G ) ≤ d + 1, and hence that ≡ ( d +1)m coincides with ≡ m . Ittherefore suffices to prove that if x ( d +1)m y, then x ( d )m y. Assume that x ( d +1)m y and let d x = d { x } ( G ) and d y = d { y } ( G ) . It isclear that d x , d y ≥ d −
1. If d x = d y = d , then our assumption that G isefficiently generated implies that x, y ∈ Frat( G ), and hence that x ≡ m y ,a contradiction. Therefore we may assume that d x = d −
1; in partic-ular G = h x, g , . . . , g d − i for some g , . . . , g d − ∈ G. If d y = d, then G = h y, g , . . . , g d − i and therefore x ( d )m y , and we are done.So assume that d x = d y = d −
1. Since x m y , without loss of generalitythere exists a maximal subgroup M of G such that x / ∈ M, y ∈ M. Let N = Core G ( M ) . Since d − ≥ ˜ m ( G ) , there exist g , . . . , g d − ∈ M such that ENERATING SETS OF FINITE GROUPS 7 h x, g , . . . , g d − i N = G. As d x = d − , we deduce from Lemma 2.11 thatthere exist n , . . . , n d − ∈ N such that G = h x, g n , . . . , g d − n d − i . On theother hand h y, g n , . . . , g d − n d − i ≤ M. Hence x ( d )m y. (cid:3) Notice that if d ( M ) < d ( G ) for every maximal subgroup M of G , then G is efficiently generated. Indeed if x / ∈ Frat( G ) , then there exists a maximalsubgroup M of G with x / ∈ M and consequently d { x } ( G ) ≤ d ( M ) < d ( G ) . But then from Lemma 2.17 we deduce the following result.
Corollary 2.18. If d ( M ) < d ( G ) for every maximal subgroup M of G , then ψ ( G ) = d ( G ) . Lemma 2.19.
Let G be a finite soluble group. If G is efficiently generatedthen ˜ m ( G ) < d ( G ) . Proof.
If suffices to prove that for every maximal subgroup M of G, wehave d ( M/ Core G ( M )) < d ( G ) = d. Assume otherwise. Then there exists amaximal subgroup M of G such that d ( M/N ) = d (where N = Core G ( M )).Furthermore, G/N = A/N : M/N and Frat( G ) ≤ N. Let a ∈ A \ Frat( G ).Then d { a } ( G ) = d, contradicting the assumption that G is efficiently gener-ated. (cid:3) The following result is now immediate from Lemmas 2.16 and 2.19.
Corollary 2.20.
Let G be a finite soluble group. Then ψ ( G ) = d ( G ) if andonly if G is efficiently generated. Theorem 2.21.
A finite soluble group G satisfies ψ ( G ) = d ( G ) if and only ifeither G is a finite p -group or there exist a finite vector space V , a nontrivialirreducible soluble subgroup H of Aut( V ) and an integer d > d ( H ) such that G/ Frat( G ) ∼ = V r ( d − : H, where r is the dimension of V over End H ( V ) and H acts in the same wayon each of the r ( d −
2) + 1 factors.Proof.
Assume that G is soluble group with ψ ( G ) = d ( G ) = d and let F = Frat( G ). By Corollary 2.20, G is efficiently generated. If N is a normalsubgroup of G properly containing F, then d ( G/N ) < d (otherwise we wouldhave d { n } ( G ) = d for every n ∈ N ). So G/F has the property that everyproper quotient can be generated by d − G/F cannot. Thegroups with this property have been studied in [8]. By [8, Theorem 1.4 andTheorem 2.7] either
G/F is an elementary abelian p -group of rank d (andconsequently G is a finite p -group) or there exist a finite vector space V and anontrivial irreducible soluble subgroup H of Aut( V ) such that d ( H ) < d and G/ Frat( G ) ∼ = V r ( d − : H, where r is the dimension of V over End H ( V ) . Conversely, if G is a finite p -group it follows immediately from Burn-side’s basis theorem that G is efficiently generated, and so ψ ( G ) = d ( G )by Corollary 2.20. Clearly a group G is efficiently generated if and only PETER J. CAMERON, ANDREA LUCCHINI, AND COLVA M. RONEY-DOUGAL if G/ Frat( G ) is efficiently generated. So to conclude the proof it suf-fices to prove that if H is a ( d − V ) and r is the dimension of V over F = End H ( V ) , then X = V r ( d − : H is efficiently generated. Notice that d ( X ) = d , so wehave to prove that d { x } ( X ) ≤ d − x = 1. Let n = r ( d −
2) + 1 . Fix a nontrivial element x = ( v , . . . , v n ) h ∈ X and let a = dim F C V ( h ) and b = n − dim F h [ V, h ] , v , . . . , v n i + dim F [ V, h ] . By [7, Lemma 5] we have d { x } ( X ) ≤ d − a + b − < r ( d − . If h = 1 , then a ≤ r − b ≤ n ; if h = 1 , then a ≤ r and b ≤ n − . In any case a + b − ≤ r + n − r + r ( d − − < r ( d − . (cid:3) Apart from p -groups, there are many examples of soluble groups that areefficiently generated. The smallest example of a soluble group which is notefficiently generated is S (we have d { x } (S ) = 2 for every x in the Kleinsubgroup): by the previous results we can conclude that ψ (S ) = 3. Problem 2.22.
Characterise the insoluble groups that are efficiently gen-erated.2.4.
Calculating ≡ m . Whilst we have not been able to determine ψ ( G ) foran arbitrary group G , we have calculated it for many small almost simplegroups G with d ( G ) = 2. It is computationally expensive to repeatedly cal-culate whether various sets of elements generates a group. In this subsectionwe describe an efficient way to calculate ≡ m - and ≡ (2)m -classes in a group,and present a theorem summarising the results of these calculations.The equivalence relation ≡ m can be thought of another way. Constructthe permutation action of G which is the disjoint union of the actions onthe cosets of maximal subgroups, one for each conjugacy class. Let Ω be thedomain of this action. For brevity, we call this the m-universal action of G . Lemma 2.23.
Let G be a finite group, and let x, y ∈ G and S ⊆ G . (1) x ≡ m y if and only if x and y have the same fixed point sets in them-universal action of G . (2) G = h S i if and only if the intersection of the fixed point sets ofelements of S in the m-universal action of G is empty.Proof. Notice that in the orbit corresponding to a non-normal maximal sub-group M , the point stabilisers are the conjugates of M ; whereas, if M isnormal, then its elements fix every point in the corresponding orbit, whilethe elements outside M fix none. Hence the fixed point set of an element x describes precisely which maximal subgroups of G contain x , and (1) follows.For (2), notice that G = h S i if and only if S is contained in no maximalsubgroup of G . (cid:3) Definition 2.24.
A permutation group action has property G if it satisfies:each set S of group elements generates the group if and only if the fixed-pointsets of elements of S have empty intersection. ENERATING SETS OF FINITE GROUPS 9
Lemma 2.25.
The m-universal action is the smallest degree permutationaction of G with property G .Proof. First notice that by Lemma 2.23(2), the m-universal action has prop-erty G . Now suppose that we have an action of G with property G . We mustshow that it contains the m-universal action. So let M be a maximal sub-group of G . Choose generators g , . . . , g r of M . Since these elements donot generate G , property G implies that they have a common fixed point,say ω . Thus M ≤ G ω , and maximality of M implies equality. So the cosetspace of M is contained in the given action. Since this holds for all maximalsubgroups M , we are done. (cid:3) Our algorithm to test whether ψ ( G ) = 2 proceeds as follows, on input afinite group G .(1) Construct the maximal subgroups of G , and hence the m-universalaction of G .(2) For each g ∈ G , compute the fixed point set Fix( g ) of g in them-universal action, and hence construct a set of equivalence classrepresentatives for the ≡ m -classes of G .(3) For each pair x, y of distinct ≡ m -class representatives, check thatthere exists a z ∈ G such that either Fix( x ) ∩ Fix( z ) = ∅ and Fix( y ) ∩ Fix( z ) is non-empty, or vice versa .If the test in Step 3 succeeds for all distinct x and y , then the set of distinct ≡ m -class representatives is also a set of distinct ≡ (2)m -class representatives.That is, ψ ( G ) = 2.We have implemented the algorithm in MAGMA [2], and used it to provethe following:
Theorem 2.26.
Let G be an almost simple group with socle of order lessthan such that all proper quotients of G are cyclic. Then ψ ( G ) = 2 . The socle of such a group G is one of: A n for 5 ≤ n ≤
7, PSL ( q ) for q ≤
27 a prime power, PSL (3), PSU (3) or the sporadic group M .The only almost simple groups with socle of order less than 10000 with aproper non-cyclic quotient are A . and PSL (25) . . Using similar ideasto the above we were able to show that ψ (A . ) = 3.Notice that in all of these instances, the lower bounds from Lemma 2.6are attained. 3. c -equivalence In this section we define another equivalence relation, which can be usedto give an easy-to-calculate upper bound on the number of ≡ m -classes, andinvestigate when this new relation coincides with ≡ m . Definition 3.1.
Let G be a finite group, and let x, y ∈ G . We define x ≡ c y if h x i = h y i . We use c for cyclic . The following is clear.
Lemma 3.2.
Let G be a finite group. For all x, y ∈ G , if x ≡ c y then x ≡ m y . Hence if n is the order of an element of G , then at least one ≡ m -class of G contains at least φ ( n ) elements. The converse implication of the first statement holds for many groups(including S n and A n for n ∈ { , } , and PSL ( q ) for q ∈ { , , } ), butnot for all groups. Proposition 3.3.
Let G be a finite group. If the relations ≡ m and ≡ c coincide, then (1) Frat( G ) = 1;(2) if G is soluble then every minimal normal subgroup of G is cyclic; (3) if G is soluble then G is metabelian.Proof. (1) All of the elements of Frat( G ) are ≡ m -equivalent.(2) Let G be soluble and let N be a minimal normal subgroup of G. Everymaximal subgroup of G either contains or complements N . This impliesthat all the elements of N \ { } are ≡ m equivalent, and consequently N iscyclic (of prime order).(3) Let G be soluble and let F = Fit( G ) . Since Frat( G ) = 1 , it follows from[24, 5.2.15] that Fit( G ) = Soc( G ), and hence F = C G ( F ) = ∩ N ∈N C G ( N ) , where N is the set of the minimal normal subgroups of G. But then GF = G T N C G ( N ) ≤ Y N Aut( N )is abelian. (cid:3) The conditions listed in the previous proposition are not sufficient toensure that the relations ≡ m and ≡ c coincide on soluble groups G . In orderto obtain a more precise result, let us fix some notation. Assume that G issoluble and satisfies the conclusions of Proposition 3.3. We set F = Fit( G )and Z = Z ( G ) . Then F = V r × · · · × V r t t × Z, where V r , . . . , V r t t are the non-central homogeneous components of F as a G -module. In particular, V i is cyclic of prime order for every i. Moreover G = F : H, where H is a subdirect product of Q i H i , with H i ≤ Aut( V i ) . Finally, for h = ( h , . . . , h t ) ∈ H, define Ω( h ) = { i ∈ { , . . . , t } | h i = 1 } . Theorem 3.4.
Let G = F : H as above be a soluble group satisfying theconclusions of Proposition 3.3. The relations ≡ m and ≡ c coincide on G ifand only if the following property is satisfied, for all ( z , h ) , ( z , h ) ∈ Z × H ( ∗ ) if h ( z , h ) i Frat H = h ( z , h ) i Frat H and Ω( h ) = Ω( h ) , then h ( z , h ) i = h ( z , h ) i . ENERATING SETS OF FINITE GROUPS 11
Proof.
Let x = ( z , h ), x = ( z , h ) ∈ Z × H, with h = ( α , . . . , α t )and h = ( β , . . . , β t ) . Assume that h x i Frat H = h x i Frat H and Ω( h ) =Ω( h ). We claim that a maximal subgroup M of G contains x if and onlyif it contains x , and hence that x ≡ m x .Let W = V r × · · · × V r t t and let L = Frat( Z × H ) = Frat( H ) . If W ≤ M, then W : L ≤ M, so h x i i ⊆ M if and only if h x i i L ⊆ M . Since h x i L = h x i L , we deduce that x ∈ M if and only if x ∈ M. If W M , thenthere exists i ∈ { , . . . , t } , a maximal H -invariant subgroup U i of V r i i and w i ∈ V r i i such that M = ( V r × · · · × V r i − i − × U i × V r i +1 i +1 × · · · × V r t t × Z ) : H w i . Notice in particular that if ( γ , . . . , γ r ) ∈ H, then ( γ , . . . , γ r ) ∈ M if andonly if γ i ∈ U i H w i i . In this case we can write γ i = u i [ w i , h − i ] h i = h i , sothat [ w i , γ − i ] ∈ U i . Since V r i i /U i ∼ = H i V i , we have that if [ w i , γ − i ] ∈ U i theneither γ i = 1 or w i ∈ U i . If w i ∈ U i then x , x ∈ M. So assume w i U i . Since Ω( h ) = Ω( h ) , we have that α i = 1 if and only only if β i = 1 , hence x ∈ M if and only if x ∈ M. We have proved that if ≡ m and ≡ c coincide,then ( ∗ ) holds.For the converse, let x = w z h , x = w z h be two elements of G with h , h ∈ H, z , z ∈ Z and w , w ∈ W. Assume that x ≡ m x . Since w h and h are conjugate in G , it is not restrictive to assume that x = z h . Weclaim that this implies that w = 1 . Indeed, assume that w = ( v , . . . , v t ) =1 . Then there exists an i such that v i = 1, and consequently there existsa maximal H -invariant subgroup U i of V r i i with v i / ∈ U i . This leads to acontradiction, since the maximal subgroup M = ( V r × · · · × V r i − i − × U i × V r i +1 i +1 × · · · × V r t t × Z ) : H contains x but not x . Having w = w = 1 , the argument used in the first part of this proofshows that the condition Ω( h ) = Ω( h ) is equivalent to saying that a max-imal subgroup of G not containing W contains x if and only if it con-tains x . On the other hand the maximal subgroups of G containing W are in bijective correspondence with those of G/ Frat H, hence the con-dition h x i Frat H = h x i Frat H is equivalent to saying that a maximalsubgroup of G containing W contains x if and only if it contains x . We have therefore proved that x ≡ m x implies that Ω( h ) = Ω( h ) and h x i Frat H = h x i Frat H, and therefore if ( ∗ ) holds, then x ≡ c x . (cid:3) Here are two examples of groups which satisfy the conclusions of Propo-sition 3.3, but do not satisfy condition ( ∗ ). Hence ≡ c -equivalence is finerthan ≡ m -equivalence.(1) Let G be the sharply 2-transitive group of degree 17, the semidirectproduct of C with a Singer cycle C . The maximal subgroupsare C : C and the conjugates of C . In particular, we see that elements of orders 2, 4 and 8 in a fixed complement C are all ≡ m -equivalent. However, ≡ c -equivalent elements have the same order.(2) A second example is ( h x i : h y i ) × h z i with | x | = 19 , | y | = 9 , | z | = 3(indeed ( y , z ) ≡ m ( y , z )). Proposition 3.5.
Assume that a finite group G contains a minimal normalsubgroup N = S × · · · × S t , with S i ∼ = S a finite nonabelian simple group.If either t ≥ , or t = 2 and S is not isomorphic to PΩ +8 ( q ) with q = 2 or , then the relations ≡ m and ≡ c do not coincide on G. Proof.
It is standard (see, for example, [1, Remark 1.1.040]) that if a maxi-mal subgroup M of G does not contain N , then one of the following occurs:(1) M ∩ N = 1;(2) M is of product type : in this case there exist α , . . . , α t ∈ Aut( S ) , independent of the choice of M , s , . . . , s t ∈ S and a proper subgroup K of S such that M ∩ N ≤ K × K s α × · · · × K s t α t ;(3) M is of diagonal type : in this case there exists a partition Φ := { B , . . . , B u } of { , . . . , t } into blocks of the same size such that M ∩ N ≤ Q B ∈ Φ D B where D B is a full diagonal subgroup of Q j ∈ B S j . By [15, Theorem 5.1] or [11, Theorem 7.1], there exist a, b ∈ S with theproperty that h a γ , b δ i = S for each choice of γ, δ ∈ S. Moreover if S =PΩ +8 ( q ) , q = 2 or 3 , then a and b are not conjugate in Aut( S ) . Let x, y ∈ S and consider g x,y = ( a x , b yα , a, . . . , a,
1) if t > a x , b yα ) otherwise.There is no maximal subgroup of product type containing g x,y . Otherwisewe would have a x ∈ K, b yα ∈ K s α , hence S = h a x , b ys − i ≤ K, contradict-ing the fact that K is a proper subgroup of S . Moreover, since either t ≥ a and b are not conjugate in Aut( S ) , no maximal subgroup of diagonaltype contains g x,y . Therefore g x,y ∈ M if and only if N ≤ M , for all maximalsubgroups M . Hence, all the elements of the subset { g x,y | x, y ∈ S } are ≡ m equivalent, and therefore the relations ≡ m and ≡ c do not coincide on G. (cid:3) Corollary 3.6.
Let G be a finite group. If the relations ≡ m and ≡ c coincideon G, then G/ Soc( G ) is soluble.Proof. Since the relations ≡ m and ≡ c coincide, Frat( G ) = 1 by Proposi-tion 3.3(1), and consequently Soc( G ) = F ∗ ( G ) , where F ∗ ( G ) is the general-ized Fitting subgroup of G .Let F ∗ ( G ) = Z ( G ) × N × · · · × N t , where N , . . . , N t are non-centralminimal normal subgroups. Since Z ( G ) = C G ( F ∗ ) = T i C G ( N i ) , we have G/Z ( G ) ≤ Q i G/C G ( N i ) . To conclude, notice that if N i is abelian, then N i is cyclic and G/C G ( N i ) is abelian, while if N i is nonabelian, then byProposition 3.5 the group N i ∼ = S t i i with t i ≤ G/ ( N i C G ( N i )) ≤ Out S ≀ Sym( t i ) , which is soluble. (cid:3) ENERATING SETS OF FINITE GROUPS 13
Problem 3.7.
Find an equivalence relation that is easier to calculate than ≡ m , but coarser than ≡ c . Determine for which insoluble groups G therelations ≡ m and ≡ c coincide.3.1. Asymptotics and enumeration.
We now briefly suggest some direc-tions for further study of the asymptotics of our new relations.
Proposition 3.8.
Let G be S n or A n . Then for almost all elements x, y ∈ G (all but a proportion tending to as n → ∞ ), the following are equivalent: (1) x ≡ m y ; (2) x ≡ (2)m y ; (3) the cycles of x and y induce the same partition of { , . . . , n } .Proof. This depends on a theorem of Luczak and Pyber [21], which statesthat for almost all x ∈ S n , the only transitive subgroups of S n containing x are S n and (possibly) A n . We restrict our attention to these elements x .Consider first the case where G = S n . Then, apart from A n , the maximalsubgroups containing x are of the form S k × S n − k , where the two orbits areunions of cycles of x . Moreover, the cycle lengths determine whether or not x ∈ A n . So (1) and (3) are equivalent.In addition, for all z ∈ G , we see that h x, z i = G whenever h x, z i istransitive, and z / ∈ A n if it happens that x ∈ A n . Membership of this set isalso determined by the cycles of x : the transitivity condition requires thatthe hypergraph whose edges are the cycles of x and z is connected. So (2)is also equivalent to (3).If G = A n , then only simple modifications are required; the argument issimpler because no parity conditions are necessary. (cid:3) Shalev in [26] proved a similar result for GL n ( q ) to Luczak and Pyber’sresult for S n : a random element of GL n ( q ) lies in no proper irreduciblesubgroup not containing SL n ( q ). This could be used to prove a similarstatement for groups lying between PSL n ( q ) and PGL n ( q ). Question 3.9.
Are there only finitely many finite almost simple groups onwhich the relations ≡ m and ≡ c coincide?Another very natural question is: how many ≡ c - and ≡ m -classes are therein the symmetric group S n ? The numbers of ≡ c -classes in the symmetricgroups S n form sequence A051625 in the On-line Encyclopedia of IntegerSequences [22]. The sequence of numbers of ≡ m classes, which begins1 , , , , , , , , . . . has recently been added to the OEIS, where it appears as Sequence A270534.If we cannot find a formula for these sequences, can we say anythingabout their asymptotics? We saw above that, for almost all elements of S n ,the ≡ m -equivalence class is determined by the cycle partition, which mightsuggest that the sequence grows like the Bell numbers (sequence A000110 in the OEIS). However, the elements not covered by this theorem can destroythis estimate.For example, let p be a prime such that the only insoluble transitivegroups of degree p are the symmetric and alternating groups. Then theabove analysis applies to all elements whose cycle type is not a single p -cycle or a fixed point and l k -cycles (where 1 + kl = p ). It is easy to showthat two elements x and y with one of these excluded cycle types satisfy x ≡ m y if and only if they satisfy x ≡ c y . So there are ( p − p -cycles, for example; this number is much greater than the p thBell number. (In this special case, we can write down a formula for thenumber of ≡ m -equivalence classes.)4. The generating graph of a group
In the remainder of the paper, we use the relations that we have definedto study an object of general interest, the generating graph of a finite group.
Definition 4.1.
The generating graph of a finite group G is the graph withvertex set G , in which two vertices x and y are joined if and only if h x, y i = G .Of course this graph is null unless G is 2-generated. We adopt the con-vention that, if the group is cyclic, then any generator of the group carriesa loop in the generating graph.A useful concept when studying the generating graph is the spread of agroup. Definition 4.2.
A group G has spread k if k is the largest number such thatfor any set S of k nonidentity elements, there exists x such that h x, s i = G for all s ∈ S .Thus the spread is nonzero if and only if no vertex of the generatinggraph except the identity is isolated; and spread at least 2 implies diameterat most 2.Among the graph-theoretic invariants which have been studied for thisgraph are the following.(1) The spread.(2) The clique number : the largest size of a set of group elements, anytwo of which generate the group.(3) The chromatic number : the smallest number of parts in a partitionof the group into subsets containing no 2-element generating set.(4) The total domination number : the smallest size of a set S with theproperty that, for any element x , there exists s ∈ S such that x and s generate the group.(5) The isomorphism type: if Γ( G ) ∼ = Γ( H ) for two groups G and H ,then when is G ∼ = H ? Definition 4.3.
In any graph X , we can define an equivalence relation ≡ Γ by the rule x ≡ Γ y if x and y have the same set of neighbours in the graph. ENERATING SETS OF FINITE GROUPS 15 (Think of Γ as meaning “graph”, or “generating” if we are thinking of thegenerating graph.) Then we define a reduced graph X whose vertices are the ≡ Γ -classes in X , two classes joined in X if their vertices are joined in X .Alternatively, we can take the vertex set to be any set of equivalence classrepresentatives, and the graph to be the induced subgraph on this set. (Theterm “reduced graph” was used by Hall [13] in his work on copolar spaces,and consequentially we term the process of producing it “reduction”; but wewarn readers that the term “graph reduction” has a very different meaningin computer science.)The reduction process preserves the graph parameters noted above: Proposition 4.4.
The clique number, chromatic number, total dominationnumber, and spread of the generating Γ( G ) are equal to the correspondingparameters of the reduced generating graph Γ( G ) . Furthermore, if Γ( G ) ∼ =Γ( H ) then Γ( G ) ∼ = Γ( H ) .Proof. Clear. (cid:3)
The following is immediate from the definition of ≡ ( r )m . Proposition 4.5.
Let G be a finite group. Then the relations ≡ Γ on Γ( G ) and ≡ (2)m on G coincide; hence ≡ m is a refinement of ≡ Γ , and is equal to ≡ Γ if and only if ψ ( G ) ≤ . Hence, in what follows, we shall write ≡ Γ to denote ≡ (2)m .Recall Definition 2.15 of efficient generation. Theorem 4.6.
Let G be a finite group with d ( G ) = 2 . (1) G has nonzero spread if and only if G is efficiently generated andhas trivial Frattini subgroup. (2) If G is soluble and has nonzero spread, then ψ ( G ) = 2 .Proof. (1) Since the spread of G is nonzero, every nonidentity element of G lies in a 2-element generating set of G , so d x ( G ) = 1 unless x = 1. Hence G is efficiently generated and Frat( G ) = 1. The converse is clear.(2) By Part (1), the assumption that G has nonzero spread implies that G is efficiently generated. Hence from Corollary 2.20, we see that ψ ( G ) = d ( G ) = 2. (cid:3) Notice that it is immediate from Theorem 4.6 that if G is a 2-generatorgroup of spread 0 and trivial Frattini subgroup, then ψ ( G ) ≥
3. For example,double transpositions are isolated vertices in Γ(S ), and so are equivalent tothe identity under ≡ Γ , though clearly not under ≡ m . In fact this group hasfourteen ≡ Γ -classes but fifteen ≡ m -classes, and as previously noted ψ (S ) =3. We shall therefore proceed for much of the following section by restrictingto groups with nonzero spread, despite that fact that we don’t know whetherTheorem 4.6(2) is also true without the solubility assumption. Conjecture 4.7.
Let G be a finite group of nonzero spread. Then ψ ( G ) ≤ G is a group with nonzero spread, then ψ ( G ) = 2whenever for all maximal subgroups M , and for all x / ∈ M , there exists z ∈ M such that h x, z i = G . This approach can be applied to S , PSL (7),and PSL (11). However, it fails in the case of A with respect to the smallestmaximal subgroups (isomorphic to S ). It also fails for PSL ( q ) for q =8 , ,
13, even though ψ ( G ) = 2 for all of these groups.5. Automorphism groups
A striking thing about generating graphs is that they have huge automor-phism groups, and these groups are poorly understood. For example, theautomorphism group of the generating graph of the alternating group A has order 2 has order 2, 3 or 5.An element of order 3 or 5 can be replaced by a nonidentity power of itselfin any generating set. Thus the sets of nonidentity powers can be permutedarbitrarily, and we find a group of order 2 (4!) = 2 of automorphismsfixing these sets. The quotient has order 120 and is isomorphic to Aut(A ) =S .Hence, for G = A , the automorphism group of the generating graph Γ( G )has a normal subgroup which is the direct product of symmetric groups onthe ≡ Γ -classes, and the quotient is the automorphism group of the reducedgraph Γ( G ). In general, a similar statement holds, but to state it we requireone further definition. Definition 5.1.
We define a weighting of the reduced generating graph, byassigning to each vertex a weight which is the cardinality of the correspond-ing ≡ Γ -class. Now let Γ w ( G ) denote the weighted graph, and let Aut(Γ w ( G ))be the group of weight-preserving automorphisms of Γ w ( G ).Note that the restriction to Aut(Γ w ( G )) is necessary, as in general anautomorphism of Γ( G ) can fail to lift to an automorphism of Γ( G ). For anexample of this, take G = PSL (16). Then Aut(Γ( G )) ∼ = 2 × Aut(PSL (16)).However, the central involution interchanges elements of order 3 with ele-ments of order 5. The ≡ m -class of the elements of order 3 has size 2, andcontains only the elements and their inverses. However, the ≡ m -class of ele-ments of order 5 has size 4 (it clearly contains all nontrivial elements of thecyclic subgroup, but in fact contains no more than this).The following theorem shows that to describe the automorphism groupof Γ( G ), it suffices to know the multiset of sizes of the ≡ Γ -classes of G , andthe automorphism group of Γ w ( G ). Theorem 5.2.
Let the ≡ Γ -classes of a finite group G be of sizes k , . . . , k n .Then A := Aut(Γ( G )) = (S k × · · · × S k n ) : Aut(Γ w ( G )) . ENERATING SETS OF FINITE GROUPS 17
Proof.
Let N := Q ni = i S k i . First we show that N ≤ A , then that A is anextension of N by a subgroup of Aut(Γ w ( G )), and finally that the whole ofAut(Γ w ( G )) is induced by A , and the extension splits.For the first claim, let x, y ∈ G such that x ≡ Γ y . Then for all z ∈ G ,there is an edge from x to z if and only if there is an edge from y to z . Hencethe map interchanging x and y and fixing all other vertices in Γ( G ) is anautomorphism of Γ( G ), so N ≤ A .For the second, we show that A acts on the ≡ Γ -classes of Γ( G ). For z ∈ G ,write N ( z ) for the set of neighbours of z in Γ( G ). Suppose that x ≡ Γ y , asbefore. Then for all a ∈ A we see that N ( x a ) = N ( x ) a = N ( y ) a = N ( y a ) , and so x a ≡ Γ y a , as required. Hence A is an extension of N by a subgroupof Aut(Γ w ( G )).For the final claim, fix an ordering of the elements in each ≡ Γ class of G ,and identify the vertices of Γ( G ) with the ordered pairs { ( i, j ) : 1 ≤ j ≤ n, ≤ i ≤ k j } . Let σ ∈ Aut(Γ w ( G )), and let j , j be adjacent verticesin Γ w ( G ), so that j σ and j σ are also adjacent. Then k j = k j σ , and for1 ≤ i ≤ k j vertex ( i, j ) is adjacent to vertex ( i, j ). Hence we can define τ to be the map sending ( i, j ) to ( i, j σ ), and then τ ∈ Aut(Γ( G )) induces σ .The result follows. (cid:3) Note that Aut( G ) preserves the generating graph Γ( G ), and hence auto-morphisms of G permute the ≡ Γ -classes. We define Aut ∗ ( G ) be the groupinduced by Aut( G ) on Γ( G ). The following is clear. Proposition 5.3.
Let G be a group with d ( G ) ≤ . Then Aut ∗ ( G ) ≤ Aut(Γ w ( G )) ≤ Aut(Γ( G )) . In the remainder of the paper we shall analyse these three automorphismgroups, concentrating on the groups G with nonzero spread. Such a group G has no non-cyclic proper quotients. Moreover (see for example [20]), itsatisfies one of the following:(1) G is cyclic;(2) G ∼ = C p × C p for some prime p ;(3) G is the semi-direct product of its unique minimal normal subgroup N (which is elementary abelian) by an irreducible subgroup C of aSinger cycle acting on N ;(4) G has a normal subgroup N ∼ = T × · · · × T r , where T , . . . , T r areisomorphic nonabelian simple groups; G/N has order rm for some m dividing | Out( T ) | , and induces a cyclic permutation of the factors.We shall show that Aut ∗ ( G ) is trivial for groups of type (1), and is equalto Aut( G ) for groups of type (3) and (4). Furthermore, we shall showthat in type (1) there is a spectacularly large gap between Aut(Γ( G )) andAut(Γ w ( G )), whilst in type (2) and (3) we find that Aut ∗ ( G ) = Aut(Γ w ( G )).First we consider the groups of type (1). Proposition 5.4.
Let G be the cyclic group of order n = p a p a · · · p a r r .Then Γ( G ) has r vertices. The group Aut ∗ ( G ) = Aut(Γ w ( G )) is trivial,while Aut(Γ( G )) ∼ = S r . Hence Aut(Γ( G )) = Q I ⊆{ ,...,r } S n I , where n I = np p · · · p r Y i ∈ I ( p i − . Proof.
First, vertices in the same coset of the Frattini subgroup Φ( G ) getidentified when we reduce the generating graph, and the weights are mul-tiplied by | Φ( G ) | = np ··· p r . So we can assume that the Frattini subgroup istrivial, that is, n = p p · · · p r .We know that in this case the ≡ Γ - and ≡ m -relations coincide, and it ismore convenient to use the latter. The group has r maximal subgroups(one of index p i for each i ) and the lattice of their intersections is the lat-tice of subsets of { , . . . , r } . So, for any subset I of { , . . . , r } , there is aunique vertex v I of the reduced graph corresponding to the intersection ofthe subgroups of index p i for i ∈ I ; and v I is joined to v J if and only if I ∩ J = ∅ .We claim that the automorphism group of Γ( G ) is the symmetric groupS r . It is clear that S r acts as automorphisms of the graph; it suffices toprove that there are no more.There is a unique vertex v ∅ joined to all others. Apart from this ver-tex, there are r vertices whose neighbour sets are maximal with respectto inclusion, namely v { i } for i = 1 , . . . , r , which must be permuted by theautomorphism group. It suffices to show that only the identity fixes allthese vertices. But any further vertex is uniquely specified by its neighbourswithin this set: v I is joined precisely to v { j } for j / ∈ I .What is the subgroup of S r fixing the weights? Recall that the weightof a vertex v I is the number of elements of G which are equivalent to thisvertex of the reduced graph, that is, which lie in the maximal subgroups ofindex p i for i ∈ I and no others. This is the number of generators of theintersection of these maximal subgroups, which is Y j / ∈ I ( p j − . Now it can happen that two of these weights are equal, even for elements inthe same S r -orbit. (For example, let n = 2 . . .
13 = 546. The subgroups oforders 2 .
13 and 3 . r preserves all the weights. Forthe minimal nonidentity elements C p i have distinct weights p i −
1, and soall are fixed by the weight-preserving subgroup. (cid:3)
Proposition 5.5.
Let G ∼ = C p . Then Γ( G ) has p +2 vertices, with Aut( G ) ∼ =GL ( p ) and Aut ∗ ( G ) ∼ = PGL ( p ) . On the other hand, Aut(Γ( G )) and Aut(Γ w ( G )) are both isomorphic to S p +1 , fixing the isolated vertex corre-sponding to the identity. Furthermore, the group Aut(Γ( G )) = S p − ≀ S p +1 . ENERATING SETS OF FINITE GROUPS 19
Proof.
Thinking of G as a vector space, two nonidentity elements x, y ∈ G fail to generate G if and only if they lie in the same 1-dimensional subspace.Furthermore, they lie in the same 1-dimensional subspace if and only if x ≡ Γ y . Thus Γ( G ) is the disjoint union of the complete graph K p +1 anda vertex representing the identity, and all weights in K p +1 are equal to p − (cid:3) Before considering the groups of type (3), we require a standard graph-theoretic definition.
Definition 5.6.
The categorical product X × Y of two graphs X and Y isthe graph whose vertex set is the cartesian product of the vertex sets, with( x , y ) joined to ( x , y ) if and only if x is joined to x in X and y isjoined to y in Y . Proposition 5.7.
Let G ∼ = C kp : C n be nonabelian with all proper quotientscyclic, and let n = p a p a · · · p a r r . The graph Γ( G ) has (2 r − p k + 2 verticesif n is squarefree, and r p k + 2 otherwise. The groups Aut( G ) and Aut ∗ ( G ) are both isomorphic to C kp : ΓL ( p k ) . Furthermore, Aut(Γ w ( G )) ∼ = S p k ,whilst Aut(Γ( G )) ∼ = S p k × S r .Proof. The elementary abelian subgroup C kp is characteristic in G , so Aut( G ) ≤ AGL k ( p ). The cyclic subgroup must embed as an irreducible subgroup of aSinger cycle, and so its centraliser in GL k ( p ) is the full Singer cycle C p k − ,and its normaliser is the normaliser of the Singer cycle, which is ΓL ( p k ).We claim that Γ( G ) is obtained from the categorical product of Γ( C n )and the complete graph K p k by the following procedure:(1) (a) If n is squarefree, identify all the vertices whose first componentcorresponds to the identity in C n .(b) Otherwise, add a vertex adjacent to all vertices whose first com-ponent corresponds to a generator in C n .The vertex in either case corresponds to the nonidentity elementsof the minimal normal subgroup of G .(2) Then add an isolated vertex corresponding to the identity.Note that generators of C n carry loops in Γ( C n ); these give rise to edges inthe categorical product between any two elements whose first componentsare equal and correspond to generators of C n .The weights of the vertices are the weights of their first components inΓ( C n ), except for the identified or added vertex in Step (1), whose weight is p k in case (1)(a) and p k ( | Φ( C n ) | −
1) in case (1)(b), and the identity whichhas weight 1.Now we demonstrate that this structure is correct.First note that in Γ( G ) all the nonidentity elements of the normal sub-group C kp are adjacent to all (and only) the generators of the complements C n ; so they all have the same neighbour sets and are ≡ Γ -equivalent. Ele-ments outside the normal subgroup are joined if and only if they lie in a different complements and their images in the C n quotient generate C n . Sotwo such elements are ≡ Γ -equivalent if they lie in the same complement andare Γ-equivalent in C n . Thus the graph has the structure claimed.We now use the results of Proposition 5.4, from which the number ofvertices of Γ( G ) follows immediately. The automorphism group of Γ( C n ) isS r , so Aut(Γ( G )) is S p k × S r .Conversely, the group Aut(Γ w ( C n )) is trivial, so the weight-preservingautomorphisms of Γ( G ) are just the permutations of the p k vertices of thecomplete graph.Finally, we prove the claims about Aut ∗ ( G ). If Aut ∗ ( G ) = Aut( G ), thenthe unique minimal normal subgroup C kp of Aut( G ) must act trivially onΓ w ( G ). However, this is not possible, for the following reason: let g be anyelement of G that generates a complement to C kp in g , and let x be anynontrivial element of C kp . Then h g i is a maximal subgroup of G , so g x
6∈ h g i and h g, g x i = G . Hence g and g x are incident in Γ( G ), and so g Γ g x .Hence x acts nontrivially on Γ( G ). (cid:3) For groups G as in the previous result, the kernel of the homomorphismfrom Aut(Γ( G )) to Aut(Γ w ( G )) is the direct product of symmetric groupswhose degrees are implicit in the proof: p k − ≡ Γ -classes in C n (which can be read off from Proposition 4.7)each p k times. The action of S p k is to permute the factors apart from theS p k − . Example 5.8.
Consider the case G = C : C . The generating graph for C = h x i is the complete graph K with the edge { , x } deleted and loopsat x and x . So the reduced graph identifies 1 and x , and also x and x , andis an edge with a loop at one end. Thus, the reduced generating graph for C : C has 12 vertices, say a , . . . , a , b , . . . , b , c, d , with all edges { a i , a j } ,all edges { a i , b j } , and no edges { b i , b j } for i = j , all edges { a i , c } , and d isolated. (Here a i corresponds to an inverse pair of elements of order 4, b i to an element of order 2, c to the four elements of order 5, and d tothe identity.) Here the kernel of the homomorphism from Aut(Γ( G )) toAut(Γ w ( G )) is S × (S ) .It remains to perform the analysis for the groups of type (4). Theorem 5.9.
Let T be a finite simple group and let N = T r ≤ G ≤ Aut( T ) ≀ h σ i , where σ acts as an r -cycle. Assume that there exists g =( y , . . . , y r ) σ , with y , . . . , y r ∈ Aut( T ) , such that G = N h g i . By substituting g by a conjugate in Aut( T ) ≀ h σ i , if necessary, we may assume that g =( y, , . . . , σ. If there exist s, t ∈ T such that T ≤ h ys, ( ys ) t i , then Aut( G ) =Aut ∗ ( G ) .Proof. Since N is the unique minimal normal subgroup of Aut( G ), if theconclusion is false, then N must act trivially on Γ( G ). But this is impossible,for the following reason. ENERATING SETS OF FINITE GROUPS 21
Let ¯ y = ys and ¯ g = (¯ y, , . . . , σ ∈ G . Notice that G contains ¯ g r =(¯ y, . . . , ¯ y ), z = ( t, , . . . ,
1) and (¯ g r ) z = (¯ y t , ¯ y, . . . , ¯ y ) . Consider the sub-group X of G generated by ¯ g and ( g r ) z . Since X contains (¯ y, . . . , ¯ y ) and(¯ y t , ¯ y, . . . , ¯ y ), we easily conclude that X = G = h ¯ g, ( g r ) z i . Now if N actstrivially, then conjugacy classes under N are contained in ≡ Γ -equivalenceclasses. Hence, in particular, ¯ g r ≡ Γ (¯ g r ) z , so G = h ¯ g, (¯ g r ) z i = h ¯ g, ¯ g r i = h ¯ g i ,a contradiction. (cid:3) Theorem 5.10.
Let G be a group of nonzero spread. Then Aut ∗ ( G ) =Aut( G ) if and only if G is nonabelian.Proof. The abelian groups of nonzero spread were considered in Proposi-tions 5.4 and 5.5, where we showed that Aut ∗ ( G ) = Aut( G ).The soluble nonabelian groups of nonzero spread were considered in Propo-sition 5.7, where we showed that Aut ∗ ( G ) = Aut( G ).The only remaining case is the insoluble groups of nonzero spread (thatis type (4)), so let G be such a group, and let N ∼ = T r = Soc( G ). We canidentify G with a subgroup of Aut( T ) ≀h σ i , where σ is the r -cycle (1 , , . . . , r ) . Let t be an involution in T and let n = ( t, , . . . , . Since G is of nonzerospread, there exists g ∈ G with G = h n, g i . Up to conjugation by an elementof (Aut T ) r , we may assume g = ( y, , . . . , σ for some y ∈ Aut( T ). Butnow G = h n, g i implies that H = h y, t i is almost simple with socle T . Since | t | = 2 , the subgroup h y, y t i is normal in H . From this we see that T ≤h y, y t i , and so by Theorem 5.9, we conclude that Aut( G ) = Aut ∗ ( G ) . (cid:3) We finish this discussion with an open problem:
Question 5.11.
Let G be an insoluble group of nonzero spread. Is Aut( G ) =Aut(Γ w ( G ))?We know of no examples where this is not the case.5.1. Calculations with Γ w ( G ) . In this subsection we describe some exper-iments that we have carried out on insoluble groups with nonzero spread.Recall the definition of the m-universal action from Subsection 2.4, andthat we showed in Theorem 2.26 that if G is almost simple, with socle oforder less than 10000 and all proper quotients cyclic then ψ ( G ) = 2. It isimmediate from Lemma 2.23(2) that two group elements x, y are incident inΓ( G ) if and only if the fixed-point sets of x and y in the m-universal actionare disjoint.For each such almost simple group G , we constructed Γ( G ) and henceAut(Γ( G )). For all such groups except for PSL (16) and PSL (25) we foundthat Aut(Γ( G )) ∼ = Aut( G ). In these remaining two cases, Aut(Γ( G )) ∼ = C × Aut( G ), but the elements in the centre of Aut(Γ( G )) do not preservethe graph weightings. From this we can conclude: Theorem 5.12.
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Peter J. Cameron, University of St Andrews, Mathematical Institute, StAndrews, Fife KY16 9SS, Scotland
E-mail address : [email protected] Andrea Lucchini, Dipartimento di Matematica, Universit`a degli studi diPadova, Via Trieste 63, 35121 Padova, Italy
E-mail address : [email protected] Colva M. Roney-Dougal, University of St Andrews, Mathematical Insti-tute, St Andrews, Fife KY16 9SS, Scotland
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