Invariable generation and the chebotarev invariant of a finite group
aa r X i v : . [ m a t h . G R ] J u l INVARIABLE GENERATION AND THE CHEBOTAREVINVARIANT OF A FINITE GROUP
W. M. KANTOR, A. LUBOTZKY, AND A. SHALEV
Abstract.
A subset S of a finite group G invariably generates G if G = h s g ( s ) | s ∈ S i for each choice of g ( s ) ∈ G, s ∈ S . We give a tight upper boundon the minimal size of an invariable generating set for an arbitrary finite group G . In response to a question in [KZ] we also bound the size of a randomlychosen set of elements of G that is likely to generate G invariably. Along theway we prove that every finite simple group is invariably generated by twoelements. Dedicated to Bob Guralnick in honor of his 60th birthday Introduction
For many years there has been a rapidly growing literature concerning the gen-eration of finite groups. This has involved the number d ( G ) of generators of agroup G , or the expected number E ( G ) of random choices of elements in order toprobably generate G , among other group-theoretic invariants. In this paper we willstudy further invariants.Dixon [Di1] began the probabilistic direction for generating (almost) simplegroups, and later he also introduced yet another direction based on the goal ofdetermining Galois groups [Di2]. This has led to the following notions: Definition.
Let G be a finite group.(a) A subset S of G invariably generates G if G = h s g ( s ) | s ∈ S i for each choiceof g ( s ) ∈ G, s ∈ S [Di2].(b) Let d I ( G ) := min (cid:8) | S | (cid:12)(cid:12) S invariably generates G (cid:9) .(c) The Chebotarev invariant C ( G ) of G is the expected value of the randomvariable n that is minimal subject to the requirement that n randomlychosen elements of G invariably generate G [KZ].There have been several papers discussing (a) for specific groups (such as finitesimple groups) [LuP, NP, Sh, FG1, KZ], but not for finite groups in general. Con-cerning (c), recall Chebotarev’s Theorem that provides elements of a suitable Galoisgroup G , where the elements are obtained only up to conjugacy in G ; the interest in(c) comes from computational group theory, where there is a need to know how longone should expect to wait in order to ensure that choices of representatives fromthe conjugacy classes provided by Chebotarev’s Theorem will generate G . This isdiscussed more carefully in [Di2, KZ]. The authors acknowledge partial support from NSF grant DMS 0753640 (W. M. K.), ERCAdvanced Grants 226135 (A. L.) and 247034 (A. S.), and ISF grant 754/08 (A. L. and A. S.). Thefirst author is grateful for the warm hospitality of the Hebrew University while this paper wasbeing written.
Our main results are the next two theorems, which depend on the classificationof the finite simple groups.
Theorem 1.1.
Every finite group G is invariably generated by at most log | G | elements. This bound is best possible: we show that d I ( G ) = log | G | if and only if G isan elementary abelian 2-group. It is trivial that d ( G ) ≤ log | G | using Lagrange’sTheorem. However, d I ( G ) may be much larger than d ( G ): Proposition 2.5 statesthat, for every r ≥ , there is a finite group G such that d ( G ) = 2 but d I ( G ) ≥ r . Theorem 3.1 contains a more precise statement of Theorem 1.1 involving the lengthand structure of a chief series of G . Theorem 1.2.
There exists an absolute constant c such that C ( G ) ≤ c | G | / (log | G | ) / for all finite groups G . This bound is close to best possible: it is easy to see that sharply 2-transitivegroups provide an infinite family of groups G for which C ( G ) ∼ | G | // (compare[KZ, Sec. 4]). In fact [KZ, Sec. 9] asks whether C ( G ) = O ( | G | / ) for all finitegroups G (which we view as rather likely).For an arbitrary finite group it is interesting to compare d I ( G ) with d ( G ), and C ( G ) with E ( G ). The upper bounds for d I ( G ) and d ( G ) are identical, although (asstated above) these quantities may be very different. On the other hand, E ( G ) ≤ ed ( G ) + 2 e log log | G | + 11 = O (log | G | ) [Lu], which is far smaller than the boundin Theorem 1.2.We will need the following result of independent interest. Theorem 1.3.
Every nonabelian finite simple group is invariably generated by elements. In fact, for proofs of Theorems 1.1 and 1.2 we will need slightly stronger resultson simple groups involving automorphisms as well (cf. Theorems 5.1 and 5.5). Thesame week that we proved these results about simple groups essentially the sameresult as Theorem 5.1 with a roughly similar proof was posted in [GM2].Dealing with simple groups uses the rather large literature of known propertiesof those groups. The fact that, for finite simple groups G , d I ( G ) and C ( G ) arebounded by some (unspecified) constant c follows for alternating groups from [LuP](cf. [KZ]), and for Lie type groups from results announced in [FG1] related to“Shalev’s ǫ -Conjecture”, which concerns the number of fixed-point-free elements insimple permutation groups (cf. Section 4).The proof of Theorem 1.2 uses bounds in [CC] and [FG1] on the number offixed-point-free elements of a transitive permutation group, together with a recentbound on the number of maximal subgroups of a finite group [LPS]. We note thatan explicit formula for C ( G ) is given in [KZ, Proposition 2.7], but we have notbeen able to use it since it appears to be too difficult to evaluate its terms for mostgroups G .The proofs of Theorems 1.1, 1.2 and 1.3 are given in Sections 3, 4 and 5, respec-tively. Section 2 contains the aforementioned result on the non-relationship of d ( G )and d I ( G ), as well as a characterization of nilpotent groups as those finite groupsall of whose generating sets invariably generate. NVARIABLE GENERATION 3
This paper is dedicated to Bob Guralnick, who has made fundamental contribu-tions in the various areas involved in this and other papers of ours.2.
Preliminary results and examples
Unless otherwise stated, we assume that the group G is finite. If X, Y ⊆ G , wesay that Y is similar to X if there is a function f : X → Y such that f ( X ) = Y and, for each x ∈ X , f ( x ) is conjugate in G to x . Thus X invariably generates G if and only if h Y i = G for each Y ⊆ G that is similar to X .Let Max( G ) denote the set of maximal subgroups of G . Let M = M ( G ) be aset of representatives of conjugacy classes of maximal subgroups of G .If M ∈ Max( G ), write f M = [ g ∈ G M g and v ( M ) = | f M || G | . Clearly f M = f M if the maximal subgroups M , M are conjugate in G . Also, f M isthe set of elements of G having at least one fixed point in the primitive permutationrepresentation of G on the set G/M of (left) cosets of M in G . Lemma 2.1.
A subset X ⊆ G generates G invariably if and only if X f M for all M ∈ M . Proof. If X ⊆ f M for some M ∈ M then each element of X is conjugate toan element of M , and hence X does not generate G invariably. Conversely, if X does not generate G invariably, then there exists a set Y similar to X such that h Y i 6 = G . Hence (using the finiteness of G ) there exist M ∈ M and g ∈ G suchthat h Y i ⊆ M g , and hence X ⊆ f M . (cid:3) The “only if” part of the above lemma also holds for infinite groups. Moreover,the proof shows that X ⊆ G generates an arbitrary group G invariably only if X e H for all H < G . This enables us to show that some infinite groups arenot invariably generated by any set of elements. For example, there are count-able groups G all of whose nontrivial elements are conjugate [HNN] (and even2-generated groups with this property [Os]), so that e H = G for every nontrivialsubgroup H and hence even G itself does not generate G invariably.However, for finite groups there are no anomalies of this kind, since e H = G forall proper subgroups H . In fact, if k ( G ) denotes the number of conjugacy classesof (elements of) the finite group G , then we have Lemma 2.2.
For any finite group G we have d I ( G ) ≤ k ( G ) . Moreover , d I ( G ) isat most the number of conjugacy classes of cyclic subgroups of G . Proof. If H is the subgroup of G generated by a set of cyclic subgroups, onefrom each conjugacy class, then the union of all conjugates of H is G , and hence H = G . (cid:3) For k ≥
1, let P I ( G, k ) be the probability that k randomly chosen elements of G generate G invariably. Lemma 2.3. max M ∈M v ( M ) k ≤ − P I ( G, k ) ≤ X M ∈M v ( M ) k . W. M. KANTOR, A. LUBOTZKY, AND A. SHALEV
Proof.
Let g , . . . , g k ∈ G be randomly chosen. Given M ∈ M , the probability that g i ∈ f M for all i is v ( M ) k . Both inequalities now follow easily from Lemma 2.1. (cid:3) We next characterize nilpotent groups in terms of invariable generation.
Proposition 2.4.
A finite group G is nilpotent if and only if every generating setof G invariably generates G . Proof.
Let Φ( G ) denote the Frattini subgroup of G . Then a subset of G generates G if and only if its image in G/ Φ( G ) generates G/ Φ( G ).Suppose G is nilpotent. Then G/ Φ( G ) is abelian. Suppose X ⊆ G generates G ,and let Y ⊆ G be similar to X . Clearly the images of X and Y in the abelian group G/ Φ( G ) coincide. Since the image of X generates G/ Φ( G ), so does the image of Y . It follows that Y generates G . We conclude that X invariably generates G .Now suppose G is not nilpotent. We shall construct a generating set X for G that does not generate G invariably using a theorem of Wielandt [Rob, p. 132]: if G/ Φ( G ) is abelian then G is nilpotent. Then G/ Φ( G ) is not abelian, and hencesome maximal subgroup M of G is not normal in G . Let g ∈ G with M g = M .Let x ∈ M g \ M and X := M ∪ { x } . Then h X i = G since M is maximal, so that M ∪ n x g − o = M is similar to X and is proper in G . This implies that X does notgenerate G invariably. (cid:3) In particular, for nilpotent G we have d I ( G ) = d ( G ). For simple groups, byTheorem 1.3 we also have the same equality (with both sides 2). However, our nextresult shows that, in general, d I ( G ) is not bounded above by any function of d ( G ): Proposition 2.5.
For every r ≥ there is a finite group G such that d ( G ) = 2 but d I ( G ) ≥ r . This group G will be a power T k of an alternating group T . For this purpose werecall an elementary criterion in [KL, Proposition 6]: Proposition 2.6.
Let G = T k for a nonbelian finite simple group T . Let S = { s , . . . , s r } ⊂ G , so that s i = ( t i , . . . , t ik ) , t ij ∈ T . Form the matrix A = t . . . t k . . .t r . . . t rk . Then S generates G if and only if the following both hold :(a) If ≤ j ≤ k then T = h t j , . . . , t rj i ; and (b) The columns of A are in different Aut( T ) -orbits for the diagonal action of Aut( T ) on T r . Proof of Proposition 2.5.
Fix n , let T = A n and let k = k ( n ) be the largest in-teger such that d ( G ) = 2, where G := G n = T k . Then k ≥ n ! / S be as in Proposition 2.6, and assume that S invariably generates G . Thenwe can arbitrarily conjugate each t ij independently and still generate G . Let C ( T )denote the set of conjugacy classes of T . Project each column β j of A to ¯ β j ∈ C ( T ) r .In view of conditions (a) and (b) in Proposition 2.6, the ¯ β j are in different Aut( T )-orbits of the diagonal action on C ( T ) r . NVARIABLE GENERATION 5
The number of conjugacy classes in T is at most c √ n , so | C ( T ) | r ≤ c r √ n . Thenumber of projections ¯ β j is k (since 1 ≤ j ≤ k ), where k ≥ n ! /
8. Then c r √ n ≥ n ! / | S | = r ≥ C √ n log n . (cid:3) Proof of Theorem 1.1
Let l ( G ) denote the length of a chief series of G . The following is a strongerversion of Theorem 1.1: Theorem 3.1.
Let G be a finite group having a chief series with a abelian chieffactors and b non-abelian chief factors. Then d I ( G ) ≤ a + 2 b. In particular , d I ( G ) ≤ l ( G ) , and if G is solvable then d I ( G ) ≤ l ( G ) . Proof.
We use induction on | G | (the case | G | = 1 being trivial). Suppose | G | > N ✁ G be a minimal normal subgroup of G . It suffices to show that d I ( G ) ≤ d I ( G/N ) + c, where c = 1 if N is abelian and c = 2 if N is non-abelian. In the latter case ourproof relies on Theorem 5.1 (proved below).Let X ⊆ G be a set of size d I ( G/N ) whose image in
G/N generates
G/N invariably.Suppose first that N is abelian. Let x ∈ N be any non-identity element of N .We claim that Y = X ∪ { x } invariably generates G . Indeed, suppose Z ⊆ G issimilar to Y . Then the image of Z in G/N generates
G/N (by the assumption on X ). Moreover, Z contains a conjugate z = x g that is a non-identity element of N . Since G/N acts irreducibly on N , h Z i ≥ N . It follows that h Z i = G , so Y generates G invariably. Thus d I ( G ) ≤ d I ( G/N ) + 1 in this case.Now suppose N is non-abelian. Then N = T × · · · × T k , where k ≥ T i are non-abelian finite simple groups such that the conjugation action of G on N induces a transitive action of G/N on the set { T , . . . , T k } .The group A := N G ( T ) /C G ( T ) is an almost simple group with socle T ⋆ := T C G ( T ) /C G ( T ) ∼ = T . By Theorem 5.1, there are elements x ∈ T ⋆ , x ∈ A suchthat h x a , x a i ≥ T ⋆ for all a , a ∈ A . Let y ∈ T , y ∈ N G ( T ), be pre-images of x , x , respectively. We claim that Y := X ∪ { y , y } invariably generates G . To see this, let Z be a set similar to Y , so Z = X ′ ∪{ y g , y g } where X ′ is similarto X and g i ∈ G ( i = 1 , Z generates G . Let K = h Z i and H = h X ′ i . Since X invariably generates G modulo N we have HN = G . Hence H acts transitively (by conjugation) on { T , . . . , T k } .Moreover, T g = T i and T g = T j for some i, j . By the transitivity of H there areelements h , h ∈ H such that T h i = T and T h j = T . Then g h , g h ∈ N G ( T ).Clearly y g h ∈ T g h = T and y g h ∈ N G ( T ) g h = N G ( T ) . Then y g h and y g h induce automorphisms of T by conjugation. In view of our choice of x and x , h y g h , y g h i induces all inner automorphisms of T . In particular, theconjugates of the element y g h ∈ T under this group generate the simple group T .Thus, K ≥ h y g h , y g h , H i ≥ T , so that K ≥ T i for all i and hence G = KN = K ,as required. W. M. KANTOR, A. LUBOTZKY, AND A. SHALEV
We see that d I ( G ) ≤ d I ( G/N ) + 2 in the non-abelian case. This completesthe proof of the first assertion in the theorem. The last two assertions followimmediately. (cid:3)
We can now complete the proof of Theorem 1.1.
Let
G, a, b be as above.Every abelian chief factor of G has order at least 2, while every non-abelian chieffactor has order at least 60. This yields | G | ≥ a b , so thatlog | G | ≥ a + (log b ≥ a + 2 b ≥ d I ( G ) , as required. Moreover, if d I ( G ) = log | G | then we must have b = 0, and all chieffactors of G have order 2. Thus G is a 2-group, so that d I ( G ) = d ( G ) = log | G | by Proposition 2.4. Now d ( G ) = log | G | easily implies that G is an elementaryabelian 2-group. (cid:3) Note that the bound in Theorem 3.1 is tight both for non-abelian simple groupsand for elementary abelian p -groups.4. Proof of Theorem 1.2
The main result of this section is the following.
Theorem 4.1.
For any ǫ > there exists c = c ( ǫ ) such that P I ( G, k ) ≥ − ǫ forany finite group G and any k ≥ c | G | / (log | G | ) / . Proof.
For M ≤ G let M G = ∩ g ∈ G M g denote the core of M in G , the kernel ofthe permutation action of G on the set of conjugates of M .Divide the set M of representatives of conjugacy classes of maximal subgroupsof G into three subsets M , M , M as follows. The set M consists of the sub-groups M ∈ M such that the primitive group G/M G is not of affine type. Theset M consists of the subgroups M ∈ M such that the primitive group G/M G is of affine type and | G : M | ≤ | G | / / (log | G | ) / . Finally, M consists of the re-maining subgroups in M , namely the subgroups M such that G/M G is affine and | G : M | > | G | / / (log | G | ) / . By [LPS, Theorem 1.3], for any finite group G we have | Max( G ) | ≤ c | G | / , where c is an absolute constant. In particular, for i = 1 , , |M i | ≤ |M| ≤ c | G | / . Fix k ≥ g , . . . , g k ∈ G be randomly chosen (we will restrict k in laterparts of the proof). By Lemma 2.1,1 − P I ( G, k ) ≤ P + P + P , where P i is the probability that g , . . . , g k ∈ f M for some M ∈ M i ( i = 1 , , k as in the statement of the theorem, P i < ǫ/ i = 1 , , P i separately. By increasing the constant c we may assume that | G | is as large as required in various parts of the proof. The set M . To bound P we use [FG1, Theorem 8.1]: the proportion of fixed-point-free permutations in a non-affine primitive group of degree n is at least c / log n , for some absolute constant c >
0. This shows that, for M ∈ M , v ( M ) ≤ − c / log | G : M | ≤ − c / log | G | . NVARIABLE GENERATION 7
By Lemma 2.3 and its proof, P ≤ X M ∈M v ( M ) k ≤ |M | (1 − c / log | G | ) k ≤ c | G | / (1 − c / log | G | ) k . Since (1 − x ) k ≤ exp( − kx ) for 0 < x <
1, for any c > log c + 3 / c log | G | − c k/ log | G | ). If k > c (log | G | ) for asuitable absolute constant c , then the latter expression tends to zero as | G | → ∞ ,and hence so does P . In particular we have P < ǫ/ | G | large enough. The set M . We next bound P . Here our main tool is the theorem that the pro-portion of fixed-point-free elements in any transitive permutation group of degree n is at least 1 /n [CC]. This implies that, if M ∈ M , then v ( M ) ≤ − | G : M | − ≤ − ( | G | / log | G | ) − / . Therefore P ≤ X M ∈M v ( M ) k ≤ |M | (cid:0) − ( | G | / log | G | ) − / (cid:1) k ≤ c | G | / (cid:0) − ( | G | / log | G | ) − / (cid:1) k . As before the right side is bounded above by exp( c log | G | − k (cid:0) | G | / log | G | ) − / ) (cid:1) for suitable c > /
2. This in turn tends to zero as | G | → ∞ for any k >c | G | / (log | G | ) / , for arbitrary c > c . Therefore P → k , and P < ǫ/ | G | . The set M . Finally we bound P . If M ∈ M then G/M G = V ⋊ H , where V is an elementary abelian p -group for some prime p , acting regularly on the set ofcosets of M in G , and H is a point-stabilizer acting irreducibly on V .Fix a chief series { G i } of G . Fix M ∈ M , and let π : G → G/M G be the canon-ical projection. The series { π ( G i ) } of normal subgroups of π ( G ) = G/M G descendsfrom G/M G = V ⋊ H to 1. If i is minimal such that π ( G i +1 ) = 1, then π ( G i ) isa minimal normal subgroup of G/M G , and hence is V , the unique minimal nor-mal subgroup of G/M G . In this situation we shall say that M uses G i /G i +1 , inwhich case G i /G i +1 ∼ = V . (For, since π ( G i ) = π ( G i ) /π ( G i +1 ) is a nontrivial G -homomorphic image of G i /G i +1 it is isomorphic to G i /G i +1 .) We have seen thatevery M ∈ M uses G i /G i +1 for a unique i . Moreover, since M ∈ M , | G i : G i +1 | = | V | = | G : M | > ( | G | / log | G | ) / . We claim that, if G is sufficiently large , then it has at most two abelian chieffactors used by any maximal subgroups in M . Indeed, if there were (at least) threesuch chief factors, appearing at places i > j > l in our chief series, then we wouldobtain the contradiction | G | ≥ | G i : G i +1 || G j : G j +1 || G l : G l +1 | > (cid:0) ( | G | / log | G | ) / (cid:1) . Fix an abelian chief factor V = G i /G i +1 of G as above. Then each g ∈ G i \ G i +1 acts fixed-point-freely on the cosets of any M that uses G i /G i +1 (since gM G ∈ V \ { } ). For each such M we have f M ⊆ G \ ( G i \ G i +1 ) . Since | G : G i | ≤ | G | / | G i : G i +1 | = | G | / | V | ≤ ( | G | log | G | ) / by the definition of M , the proportion of elements g ∈ G i \ G i +1 inside G is atleast | G : G i | − ≥ ( | G | log | G | ) − / . Since the union of f M k over all M using G i /G i +1 is contained in (cid:0) G \ ( G i \ G i +1 ) (cid:1) k , it follows that the probability that W. M. KANTOR, A. LUBOTZKY, AND A. SHALEV randomly chosen elements g , . . . , g k of G all lie in f M for some such M is at most(1 − ( | G | log | G | ) − / ) k . Although there may be many choices for M in M , thereare at most two choices for the chief factor G i /G i +1 . Thus, P ≤ (cid:0) −
12 ( | G | log | G | ) − / (cid:1) k ≤ (cid:0) − k | G | log | G | ) − / (cid:1) , where the right hand side is less than ǫ/ k ≥ c ( | G | log | G | ) / for some c = c ( ǫ ).Our bounds on the three probabilities P i complete the proof. (cid:3) Remark.
Recall that the ǫ -conjecture , posed by the third author of this paper,states that there exists an absolute constant ǫ > ǫ . This amounts to saying that v ( M ) ≤ − ǫ for any finite simple group G and any M ∈ Max( G ). This conjecture holds for alternating groups [LuP] andfor Lie type groups of bounded rank [FG1, Secs. 3 and 4]. Moreover, in [FG1,Theorem 1.3] it is announced that the ǫ -conjecture holds in general, and proofs insome additional cases appear in [FG2]. When M ∈ M our proof of Theorem 4.1uses [FG1, Theorem 8.1], which in turn relies on the ǫ -conjecture. However, we nowshow that Theorem 5.5 below easily yields a weaker version of [FG1, Theorem 8.1]that still suffices for our purpose. The set M revisited. Namely, we claim that there exists c > v ( M ) ≤ − c (log | G | ) − | G | − / , where G is any non-affine primitive permutation group and M is a point-stabilizer. For, if s , s generate G invariably, and if M ∈ Max( G ) , then f M ∩ s Gi = ∅ for i = 1 or 2, in which case v ( M ) ≤ − | s Gi | / | G | . Then v ( M ) ≤ − | G | − / foreach sufficiently large finite simple group G and each such M , by Theorem 5.5.This implies that, for all finite simple groups G and all M ∈ Max( G ), we have v ( M ) ≤ − c | G | − / for some constant c > G is an almost simple group with socle T then, since | Out( T ) | ≤ c log | T | (cf. [GLS, Sec. 2.5]), we easily obtain v ( M ) ≤ − c (log | G | ) − | G | − / for all M ∈ Max( G ) not containing T , for some c >
0. Our claim follows bycombining this inequality with the reduction to almost simple groups given in theproof of [FG1, Theorem 8.1].Thus, if M ∈ M , then the above claim yields P ≤ X M ∈M v ( M ) k ≤ c log | G | (1 − c (log | G | ) − | G | − / ) k . The right hand side tends to zero when k ≥ c (log | G | ) | G | / ; but for the proofof Theorem 4.1 we can assume the stronger inequality k ≥ c | G | / (log | G | ) / .Consequently P →
0, as required.
Completion of proof of Theorem 1.2 . Apply Theorem 4.1 with ǫ = 1 / c = c (1 / k = ⌈ c | G | / (log | G | ) / ⌉ . Then k randomly chosen elements of G invariably generate G with probability at least 1 /
2. This implies that C ( G ) ≤ k ≤ (2 c + 1) | G | / (log | G | ) / . (cid:3) NVARIABLE GENERATION 9
Corollary 4.2. (a) If G is a finite group without abelian composition factors , then C ( G ) = O ((log | G | ) ) . (b) If G is an almost simple group , then C ( G ) = O (log | G | log log | G | ) . Proof.
We have already seen (a) in our first treatment of the non-affine case ( M ∈M ) of Theorem 4.1.To prove (b) we first note that, for some c > M ∈ M , we have v ( M ) ≤ − c/ log | G | . Indeed, if M has trivial core then this follows from [FG1, Theorem8.1] (and hence from the correctness of the ǫ -conjecture stated above). Otherwise, M contains the simple socle T of G , and | G/T | ≤ |
Out( T ) | ≤ c log | T | ≤ c log | G | as noted above. In this situation, if g ∈ G acts fixed-point-freely on the cosets of M in G , so do all the elements of gT , so that v ( M ) ≤ − c − / log | G | .By [GLT, Theorem 1.3], |M| ≤ c (log | G | ) when G is almost simple. This yields X M ∈M v ( M ) k ≤ c (log | G | ) (1 − c/ log | G | ) k ≤ c (log | G | ) exp( − ck/ log | G | ) . The right hand side tends to zero as | G | → ∞ when k ≥ c log | G | log log | G | . Thisproves part (b). (cid:3) We observe that the bound in (b) is almost best possible, up to the log log | G | factor. To show this we use the following example [FG1, p. 115]. Fix any prime p . Let G = PSL(2 , p b ) .b , the extension of the simple group by the group B of b field automorphisms, where b is a prime not dividing p ( p − G act on thecosets of the maximal subgroup N G ( B ) of G . Then all fixed-point-free elementsare contained in the socle of G , so their proportion is less than 1 /b . Therefore v ( M ) ≥ − /b .Hence, by Lemma 2.3, P I ( G, k ) ≤ − (1 − /b ) k , so that for sufficiently large b we obtain P I ( G, k ) ≤ − (1 − c / log | G | ) k ≤ − exp( − c k/ log | G | ) , where c , c are suitable constants. Thus P I ( G, k ) ≤ / k ≤ c log | G | , where c > k + 1 randomchoices of elements to invariably generate G is 1 − P I ( G, k ). By the definition ofthe expectancy C ( G ) we have C ( G ) ≥ ( k + 1)(1 − P I ( G, k )). If k = [ c log | G | ]then 1 − P I ( G, k ) ≥ / k + 1 ≥ c log | G | . This yields C ( G ) ≥ ( k + 1)(1 / ≥ ( c /
2) log | G | . Simple groups
We will prove the following slightly stronger version of Theorem 1.3:
Theorem 5.1.
Let G be a finite simple group. (a) If G is not one of the groups PΩ + (8 , q ) , q = 2 or , then there are twoelements s , s ∈ G such that G = h s g , s g i for each choice of g i ∈ Aut( G ) . (b) If G is PΩ + (8 , q ) , q = 2 or , and if G ≤ G ⋆ ≤ Aut( G ) , then there areelements s ∈ G, s ∈ G ⋆ such that G ≤ h s g , s g i for each choice of g i ∈ G ⋆ . Of course, Theorem 1.3 is just (a) using inner automorphisms. This theoremis also obtained in [GM2, Theorem 7.1], along with the fact that PΩ + (8 ,
2) is anactual exception.We begin with the easiest case:
Table 1.
Classical groups quasisimple G | t | t on V | t | t on V SL( n,q ) ( q n − / ( q − n ( q n − − / ( q −
1) ( n − ⊕ n oddSL( n,q ) ( q n − − / ( q −
1) ( n − ⊕ q n − / ( q − nn ≥ m, q ) q m + 1 2 m lcm( q m − + 1 , q + 1) (2 m − ⊥ m ≥ m + 1 , q ) ( q m + 1) / m − ⊥ q m − / m ⊕ m ) ⊥ q oddΩ + (4 k, q ) ( q n ′ − + 1) /δ ( n − − ⊥ − lcm( q n ′ − +1 , q +1) /δ ( n − − ⊥ − n = 2 n ′ = 4 k Ω + (4 k + 2 , q ) ( q n ′ − + 1) /δ ( n − − ⊥ − ( q n ′ − /δ n ′ ⊕ n ′ n ′ = 4 k + 2Ω − (4 k, q ) ( q n ′ + 1) /δ n − ( q n ′ − − /δ ( n − + ⊥ − n = 2 n ′ = 4 k Ω − (4 k + 2 , q ) ( q k +1 + 1) /δ (4 k + 2) − ( q k + 1) /δ k − ⊥ + SU(2 m, q ) q m − + 1 (2 m − ⊥ q m − / ( q + 1) 2 m SU(2 m + 1 , q ) ( q n + 1) / ( q + 1) n q n − − n − ⊥ Lemma 5.2.
Theorem 5.1 holds for each alternating group A n , n ≥ . Proof. If n = 6 then Aut( A n ) = S n . For even n > n − p -cycle and ( n − p )-cyclefor a prime p ≤ n − n ; it is easy to check that such a prime exists.These two elements generate a group H that is readily seen to be transitive andeven primitive. Since H contains a p -cycle, H = A n by a classical result of Jordan[Wie, Theorem 13.9].If n is odd then an n -cycle and a p -cycle can be used in the same manner, foran odd prime p ≤ n − n .Finally, A is generated by any elements of order 4 and 5. (cid:3) For groups of Lie type we will use the knowledge of all maximal overgroups M of a carefully chosen semisimple element t . Then, by Lemma 2.1, we only need tochoose an Aut( G )-conjugacy class of elements that does not meet the union of thecorresponding sets f M . Our arguments differ from those in [GM2] primarily due tothat paper using [GM1] whereas we rely more on the earlier paper [MSW]. Lemma 5.3.
Theorem 5.1 holds for each classical simple group other than PΩ + (8 , q ) . Proof.
We will consider the corresponding quasisimple linear group G , usingsemisimple elements t and t in Table 1 that decompose the space as indicated inthe table. (Here δ i is 1 or 2, n is the dimension of the underlying vector space V ,and n ′ = n/
2. If an entry involves lcm( q i + 1 , q j + 1) for some i, j , then t inducesirreducible elements of order q i +1 or q j +1 on the indicated subspaces of dimension2 i or 2 j .)In each case, t is the element called “ s ” in [MSW, Theorem 1.1]; if there isa 1 − or 2 − space indicated then it is centralized. For each group G , all maximalovergroups of t are listed in [MSW, Theorem 1.1]. Until the end of the proof we will NVARIABLE GENERATION 11 exclude the case G = Sp(4 , q ). Then all automorphisms of G act on V , preservingthe underlying geometry [GLS, Sec. 2.5]. It follows that all Aut( G )-conjugates of t i act on V as t i does (for i = 1 , t and t that haveno assumed relationship to one another, so if the two elements studied generate G then they invariably generate G .If G is not SL(2 , q ), Sp(4 , q ) or Sp(8 , t and t invariably generate G by[MSW, Theorem 1.1]: all of the exceptions in that theorem do not arise here due tothe behavior of both t and t on V . If G = Sp(8 ,
2) then we replace t by anotherelement, as follows. Let f ∈ G have order 5 and centralize a nondegenerate 4 − space.Then C G ( f ) = h f i × Sp(4 , c = (1 , , , , ∈ S ∼ = Sp(4 , < C G ( f ).Then c / ∈ S ∼ = O − (4 , f c is not in an overgroup O − (8 ,
2) of t . Sinceits order implies that f c is also not in any of the other maximal overgroups of t [MSW, Theorem 1.1], it follows that t and f c invariably generate G .Case SL(2 , q ). When q is 4 , q = 7, elements oforder 7 and 4 invariably generate G . For all other q ≥
4, the same t and t asindicated in the table (but with t acting irreducibly on each 1 − space) invariablygenerate G by [Di, Ch. XII].Case Sp(4 , q ). We may assume that q ≥ ,
2) is not simple andPSp(4 , ∼ = PSU(4 , t and t as in the table, such that t in-duces an element of order q + 1 inside the Sp(2 , q ) produced by each factor in thedecomposition 4 = 2 ⊥
2. Once again t and t invariably generate G by [MSW,Theorem 1.1]. (cid:3) We note that classical groups were considered in [NP, Section 10] from a prob-abilistic point of view: a large number of pairs of elements was described thatinvariably generate various classical groups. The group GL( n, q ) was also handledin [Sh] for large n . All groups of Lie type also were dealt with probabilistically, atleast for bounded rank, in [FG1, Theorem 5.3]. Lemma 5.4.
Theorem 5.1 holds for PΩ + (8 , q ) . Proof.
Once again we will consider the corresponding linear group G = Ω + (8 , q ),using the properties of Aut( G/Z ( G )) contained in [GLS, Sec. 2.5]. We have G/Z ( G ) ≤ G ⋆ ≤ Aut(
G/Z ( G )).(a) Suppose first that q >
3. We will use the same h t i as above (mod Z ( G )),of order ( q + 1) / (2 , q − + = 6 − ⊥ − , centralizing the2 − space.We also use an element t ∈ G of order ( q − / (2 , q − t decomposesour space as 8 + = (3 ⊕ ⊥ (1 ⊕
1) using totally singular 3 − and 1 − spaces,inducing isometries of order q − ⊕ q − ⊕
3, and hence acting irreducibly on the indicated 3 − spaces. Then t fixes exactly two singular 1 − spaces, and two totally singular 4 − spaces in each G -orbit of such 4-spaces (each of the latter fixed subspaces has the form 3 ⊥ τ is any automorphism of G/Z ( G ), then t τ has the same properties. In particular,neither t nor t τ fixes any anisotropic 1 − or 2 − space for any τ ∈ Aut(
G/Z ( G )).(N. B.–This requires that q >
3: if q = 3 then the analogous element t induces − + − space 1 ⊕ − spaces.) However, by [MSW,Theorem 1.1] each maximal subgroup of G/Z ( G ) that contains t (mod Z ( G )) eitherfixes such a 1 − or 2 − space or its image under a triality automorphism behaves that way. Hence, there is no maximal subgroup containing t and t mod Z ( G ), and wehave invariably generated G/Z ( G ).(b) From now on q ≤
3. First consider the case where G ⋆ acts (projectively) on V (this includes the situation in Theorem 1.3). We use elements t and t of G/Z ( G )of order ( q − / (4 , q −
1) arising from a decomposition 8 + = 4 − ⊥ − ⊥ + andfrom a decomposition 8 + = 4 ⊕ − spaces (the correspondingcyclic groups h t i i are conjugate under Aut( G/Z ( G )) but not under G ⋆ ). The Sylow5-subgroups of h t i and h t i behave differently on the vector space, and h t i is anelement of order ( q − / (4 , q −
1) that acts fixed-point-freely on V . Hence, by[Kl], h t , t i is contained in no proper subgroup of G , so that h t , t i = G .Finally, suppose that G ⋆ does not have any Aut( G/Z ( G ))-conjugate that acts on V . Here we return to the original setting of the theorem, now letting G denote thesimple group PΩ + (8 , q ). Since Out( G ) ∼ = S or S , we may assume that G ⋆ containsa triality outer automorphism. Consequently, there is a subgroup Z × SL(3 , q ) of G ⋆ that contains an element t of order 3( q + q + 1) such that τ = t q + q +15 is atriality automorphism and t acts projectively on V as 8 + = (3 ⊕ ⊥ (1 ⊕ h t , t i ∩ G ≥ h t , t i is either G , Ω(7 , q ) or lies in A < Ω + (8 , h t , t i ∩ G is invariant under τ , only the first of these canoccur (for example, h t , t i ∩ G cannot be A or PSL(2 , < A ). Thus, t and t invariably generate G h τ i . (cid:3) Completion of proof.
In [GM1, Tables 6 and 9] there are lists of carefully chosencyclic subgroups of exceptional and sporadic simple groups, as well as all of themaximal overgroups M of those subgroups. It is straightforward to use those tablesto handle these final cases of Theorem 5.1. This amounts to exhibiting an elementorder for G not appearing in any of the listed subgroups M . We provide somedetails for the exceptional groups. Table 2 reproduces part of [GM1, Table 6]. Here T is a cyclic maximal torus and M runs through the isomorphism types of maximalovergroups of T . (Notation: ǫ = ±
1, Φ n = Φ n ( q ) is the n th cyclotomic polynomialevaluated at q , Φ ′ = Φ ′ ( q ) = q + √ q + 1, Φ ′ = Φ ′ ( q ) = q + √ q + 1 andΦ ′ = Φ ′ ( q ) = q + √ q + q + √ q + 1.) In each case, the order of t guaranteesthat it is not contained in any of the listed maximal overgroups M (there are alsoother choices for t ). Hence, a generator of T together with t behave as requiredin the theorem. (cid:3) In Section 4 we needed a bit more information than in the preceding theoremfor an alternative proof of Theorem 4.1 and hence of Theorem 1.2:
Theorem 5.5.
For all sufficiently large G in Theorem 5.1 , the elements s i can bechosen so that | s Gi | > | G | / / for i = 1 , . Proof.
This is a straightforward matter of examining each part of the proof ofTheorem 5.1. In each case we need to check that | C G ( s i ) | < | G | / for i = 1 , | G | .For alternating groups, when n is even each of the groups C G ( s ) is the directproduct of two cyclic groups, and hence has order satisfying the required bound.When n is odd the same holds if we replace the p -cycle by the product of a disjoint p -cycle and an ( n − p )-cycle (a power of which is a p -cycle).In Lemma 5.3 – excluding SL(2 , q ) – we have | C G ( T ) | ∼ q r and | C G ( t ) | ∼ q r ,where r is the rank of the corresponding algebraic group. (For example, for SL( n, q ) NVARIABLE GENERATION 13
Table 2.
Exceptional groups G | T | M ≥ T further max. | t | B ( q ) Φ ′ N G ( T ) − Φ ′ ( − q ) q ≥ G ( q ) Φ ′ N G ( T ) − Φ ′ ( − q ) q ≥ G ( q ) , | q + ǫ q + ǫq + 1 SL ǫ (3 , q ) . , q − ǫq + 1( q = 4) G ( q ) , | q q + q + 1 SL(3 , q ) . , q − q + 1( q = 3) D ( q ) Φ N G ( T ) − ( q +1)( q − / (2 , q − F ( q ) Φ ′ N G ( T ) − Φ ′ ( − q ) q ≥ F ( q ) Φ
12 3 D ( q ) . , . , q + 1 F (2) ( q =2) , PSL(4 , . ( q =2) E ( q ) Φ / (3 , q −
1) SL(3 , q ) . − ( q +1)( q − / (6 , q − E ( q ) Φ / (3 , q + 1) SU(3 , q ) . − ( q − q +1) / (6 , q +1) E ( q ) Φ Φ / (2 , q − E ( q ) sc .D q +1 − Φ / (2 , q − E ( q ) Φ N G ( T ) − Φ we have | C G ( T ) | = ( q n − / ( q −
1) or q n − −
1, for Sp(2 m, q ) we have | C G ( t ) | ≤ ( q m − + 1)( q + 1), and for Ω + (4 k + 2 , q ) we have | C G ( T ) | ≤ ( q k + 1)( q + 1).) Astraightforward calculation using | G | verifies that these bounds are small enough forour purposes. When G = SL(2 , q ) we have | C G ( T ) | = q + 1, so that | s Gi | > | G | / / (cid:3) Random generation.
We conclude with remarks concerning the random gen-eration of finite simple groups. All finite simple groups G are generated by tworandomly chosen elements with probability tending to 1 as | G | → ∞ [Di1, KL, LS].We claim that this does not hold for invariable generation: the probability that two –or any bounded number of – random elements of a finite simple group G invariablygenerate G is bounded away from . To show this we need the following result thatis implicit in [FG1].
Lemma 5.6.
There exists an absolute constant ǫ > such that any finite simplegroup G has a maximal subgroup M for which v ( M ) ≥ ǫ . Proof.
This is trivial for alternating groups A n , where we take M to be a point-stabilizer in the natural action, so v ( M ) ∼ − e − . For groups G of Lie type ofbounded rank over a field with q elements we may assume q is large, and then theresult follows with M a maximal subgroup containing a maximal torus (see thediscussion in [FG1, start of Sec. 4]). For classical groups of large rank the resultfollows from [FG1, Theorem 1.7]. Sporadic simple groups satisfy the conclusiontrivially. (cid:3) This lemma can be considered as a kind of weak analogue of the ǫ -conjecture(stated above) but in the opposite direction. We can now deduce
Corollary 5.7.
There is an absolute constant ǫ > such that P I ( G, k ) ≤ − ǫ k for all finite simple groups G and positive integers k . Proof.
This follows by combining the above lemma with Lemma 2.3. (cid:3)
In [FG1, p. 114] it is announced that, for any ǫ >
0, there is c = c ( ǫ ) such that P I ( G, k ) ≥ − ǫ whenever G is a finite simple group of Lie type and k ≥ c . Thecase of bounded rank is proved in [FG1, Theorem 4.4], and a similar result foralternating groups was proved earlier in [LuP].Using these results it follows that, for any function f : N → N such that f ( n ) → ∞ as n → ∞ (even if arbitrarily slowly), we have P I ( G, f ( | G | )) → G whose orders tend to infinity. References [CC] P. J. Cameron and A. M. Cohen, On the number of fixed point free elements in apermutation group, Discrete Math. 106/107 (1992) 135–138.[Di] L. E. Dickson, Linear groups with an exposition of the Galois field theory. Dover (reprint),New York 1958.[Di1] J. D. Dixon, The probability of generating the symmetric group. Math. Z. 110 (1969)199–205.[Di2] J. D. Dixon, Random sets which invariably generate the symmetric group. Discrete Math.105 (1992) 25–39.[FG1] J. Fulman and R. M. Guralnick, Derangements in simple and primitive groups. Groups,combinatorics & geometry (Durham, 2001; Eds. A. A. Ivanov, M. W. Liebeck andJ. Saxl), 99–121, World Sci. Publ., River Edge, NJ 2003.[FG2] J. Fulman and R. M. Guralnick, Bounds on the number and sizes of conjugacy classesin finite Chevalley groups with applications to derangements (to appear in Trans. AMS;preprint arXiv:0902.2238v1).[GLS] D. Gorenstein, R. Lyons and R. Solomon, The classification of the finite simple groups.Number 3. Part I. Chapter A. Almost simple K-groups. AMS, Providence 1998.[GLT] R. M. Guralnick, M. Larsen and P. H. Tiep, Representation growth in positive charac-teristic and conjugacy classes of maximal subgroups (preprint arXiv:1009.2437).[GM1] R. M. Guralnick and G. Malle, Products of conjugacy classes and fixed point spaces(preprint arXiv:1005.3756v2).[GM2] R. M. Guralnick and G. Malle, Simple groups admit Beauville structures (preprintarXiv:1009.6183).[HNN] G. Higman, B. H. Neumann and H. Neumann, Embedding theorems for groups. J. Lon-don Math. Soc. 24 (1949) 247–254.[Kl] P. B. Kleidman, The maximal subgroups of the finite 8-dimensional orthogonal groups P Ω +8 ( q ) and of their automorphism groups. J. Algebra 110 (1987) 173–242.[KL] W. M. Kantor and A. Lubotzky, The probability of generating a finite classical group.Geom. Ded. 36 (1990) 67–87.[KZ] E. Kowalski and D. Zywina, The Chebotarev invariant of a finite group (preprintarXiv:1008.4909v).[LPS] M. W. Liebeck, L. Pyber and A. Shalev, On a conjecture of G. E. Wall. J. Algebra 317(2007) 184–197.[LS] M. W. Liebeck and A. Shalev, The probability of generating a finite simple group. Geom.Ded. 56 (1995) 103–113.[Lu] A. Lubotzky, The expected number of random elements to generate a finite group. J.Algebra 257 (2002) 452–459.[LuP] T. Luczak and L. Pyber, On random generation of the symmetric group. Combin.Probab. Comput. 2 (1993) 505–512.[MSW] G. Malle, J. Saxl and T. Weigel, Generation of classical groups. Geom. Ded. 49 (1994)85–116. NVARIABLE GENERATION 15 [NP] A. Niemeyer and C. E. Praeger, A recognition algorithm for classical groups over finitefields. Proc. London Math. Soc. 77 (1998) 117–169.[Os] D. Osin, Small cancellations over relatively hyperbolic groups and embedding theorems.Ann. Math. 172 (2010) 1–39[Rob] D. J. Robinson, A Course in the Theory of Groups. Springer, New York 1982.[Sh] A. Shalev, A theorem on random matrices and some applications. J. Algebra 199 (1998)124–141.[Wie] H. Wielandt, Finite Permutation Groups. Academic Press, New York and London 1964.
University of Oregon, Eugene, OR 97403
E-mail address : [email protected] Institute of Mathematics, Hebrew University, Jerusalem 91904
E-mail address : [email protected] Institute of Mathematics, Hebrew University, Jerusalem 91904
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