Characteristic Covering Numbers of Finite Simple Groups
aa r X i v : . [ m a t h . G R ] J a n CHARACTERISTIC COVERING NUMBERS OF FINITE SIMPLEGROUPS
MICHAEL LARSEN, ANER SHALEV, AND PHAM HUU TIEP
Abstract.
We show that, if w , . . . , w are words which are not an identity ofany (non-abelian) finite simple group, then w ( G ) w ( G ) · · · w ( G ) = G for all (non-abelian) finite simple groups G . In particular, for every word w , either w ( G ) = G for all finite simple groups, or w ( G ) = 1 for some finite simple groups.These theorems follow from more general results we obtain on characteristic collec-tions of finite groups and their covering numbers, which are of independent interestand have additional applications. Introduction
The theory of word maps on groups, and on finite simple groups in particular, datesback to Borel [Bo], and has developed significantly in the past 3 decades; see for instance[MZ], [SW], [LiSh1], [La], [Sh1], [LS1], [LS2], [LOST], [LST1], [GT], as well as themonograph [Se] and the survey paper [Sh2].Most of these works are of asymptotic nature. For example, it is shown in [LST1](following [LS1, LS2]) that, for every non-identity word w there exists N ( w ) ∈ N suchthat, if G is a finite simple group of order at least N ( w ) then w ( G ) = G . Recall that a word is an element of a free group F d of rank d , and it defines a word map w : G d → G for every group G , induced by substitution. The image of this word map is denoted by w ( G ).Here and throughout this paper, by a finite simple group we mean a non-abelianfinite simple group.Non-asymptotic results on word maps are often very challenging and require moretools. See, for instance, [LOST], [LOST2], [GM], [GT] and [GLOST]. In [LOST] itis shown that the commutator map is surjective on all finite simple groups, provingthe Ore Conjecture. Using a somewhat similar strategy, it was subsequently proved in[LOST2] that, if w = x p for a prime p = 3 ,
5, then w ( G ) = G for all finite simplegroups G . In [GM] a stronger result is established, yielding the same conclusion for w = x k , where k is any prime power or a power of 6. Mathematics Subject Classification.
Primary 20D06.ML was partially supported by NSF grant DMS-2001349. AS was partially supported by ISF grant686/17 and the Vinik Chair of mathematics which he holds. PT was partially supported by NSF grantDMS-1840702, the Joshua Barlaz Chair in Mathematics, and the Charles Simonyi Endowment at theInstitute for Advanced Study (Princeton). The authors were also partially supported by BSF grant2016072.The authors are grateful to Frank L¨ubeck for kindly providing us with the character table of theSteinberg group D (3). The above result was further extended in [GLOST]. Consider the power words w = x k and w = x ℓ , where k = p a q b for primes p, q and ℓ is an odd integer. By celebratedtheorems of Burnside and of Feit and Thompson, finite groups satisfying the identity w or the identity w are solvable. In particular, w , w are not identities of any finitesimple group. It was proved in [GLOST] that, w ( G ) = G and w ( G ) = G for all finite simple groups G .Our main theorems, stated below, may be regarded as far-reaching extensions of theabove mentioned results. In particular, we derive similar conclusions for every word w which is not an identity of any finite simple group, sometimes with somewhat largerexponents. Our strategy is to pose and study a more general problem. Definition 1.1.
A collection of non-empty subsets S ( G ) ⊆ G , one for each finite group G , is characteristic if for every homomorphism ϕ : G → H we have ϕ ( S ( G )) ⊆ S ( H ).Some comments are in order.1. For any characteristic collection S and for any finite group G , S ( G ) must be afully invariant – in particular, a characteristic – subset of G (take ϕ : G → G );furthermore, 1 ∈ S ( G ) (take ϕ to be the trivial endomorphism of G ).2. Trivial examples of characteristic collections are S ( G ) = { } for all G (theminimal collection) and S ( G ) = G for all G (the maximal collection).3. Given characteristic collections S, T , define their product ST by ST ( G ) := S ( G ) T ( G ), which is easily seen to be a characteristic collection. This binaryoperation is associative with an identity element (the minimal collection). Thusthe set of all characteristic collections is a monoid, which is partially ordered byinclusion (where S ⊆ T if S ( G ) ⊆ T ( G ) for all G ).4. Let w ∈ F d be a word, and consider the word maps it defines on finite groups G . Then the collection S ( G ) := w ( G ) is obviously characteristic.5. Let P d denote the free profinite group on x , . . . , x d (namely the profinite com-pletion of F d ). For any finite group G and g = ( g , . . . , g d ) ∈ G d there is aunique homomorphism ψ g : P d → G satisfying ψ g ( x i ) = g i ( i = 1 , . . . , d ). Anyelement W ∈ P d (which may be regarded as a profinite word) gives rise to afunction (a profinite word map) W : G d → G satisfying W ( g ) := ψ g ( W ). Set-ting S ( G ) := W ( G ), the image of W , we obtain a characteristic collection S .It is easy to see that different elements W , W ∈ P d give rise to distinct char-acteristic collections S , S . Hence the monoid of characteristic collections hascardinality 2 ℵ .6. Any word w in one variable, i.e. w = x k for some integer k , defines a wordmap w : G → G on each finite group G , and the collection of kernels S ( G ) := w − (1) = { g ∈ G : g k = 1 } is characteristic. Definition 1.2.
A characteristic collection S is ample if | S ( G ) | ≥ G .For example, let w be a word, and let S be its associated characteristic collection(i.e. S ( G ) = w ( G )). Then the ampleness of w is equivalent to each of the followingconditions.(a) w is not an identity of any finite simple group. HARACTERISTIC COVERING NUMBERS OF FINITE SIMPLE GROUPS 3 (b) w is not an identity of any minimal finite simple group (these are certain well knowngroups of type PSL or Suzuki type).(c) For all finite groups G , w ( G ) = 1 implies that G is solvable. Definition 1.3.
We define the characteristic covering number ccn ( G ) of a finite group G to be the smallest integer n such that if S , . . . , S n are ample characteristic collections,then S ( G ) · · · S n ( G ) = G . If no such n exists, we say ccn ( G ) = ∞ .The main result of this paper is the following: Theorem A. If G is a finite simple group, then ccn ( G ) ≤ Theorem B. If w , . . . , w are words in disjoint letters, and none of the w i is an identityon any finite simple group, then the juxtaposition w · · · w is surjective on every finitesimple group. Corollary C. If w is a word which is not an identity on any finite simple group, then w ( G ) = G for every finite simple group G .Clearly, Theorem B and Corollary C also follow when w, w i are replaced by profinitewords W , W i .Note that the condition that none of the w i (or w ) is an identity on any finite simplegroup in Theorem B and Corollary C is necessary. As shown in [KaN], [GT], for anyinteger N , there exist a word w and a finite simple group G such that w ( G ) = 1 and w ( G ) N = G .As for lower bounds, it follows from [GLOST, 8.9] that ccn ( G ) ≥ G ; indeed, there are odd integers ℓ and finite simple groups G such that w ( G ) = G for w = x ℓ . This can be improved as follows. In the special case that w = x , the collection S ( G ) = w − (1) is ample by Feit-Thompson. The minimal n such that S ( G ) n = G is the involution width of G , denoted iw ( G ). This invariant hasbeen investigated by several mathematicians (see [Ma] and the references therein). Inparticular, Kn¨uppel and Nielsen showed [KnN, 16] that iw (SL n ( q )) ≥ n ≥ q ≥
7. When, in addition, gcd( n, q −
1) = 1, we have SL n ( q ) = PSL n ( q ), and thisgives a lower bound of 4 for the characteristic covering number of infinitely many finitesimple groups.It would be interesting to know whether our upper bound of 6 for ccn can be improved.Malcolm showed [Ma] that iw ( G ) ≤ ccn as well. For most finite simple groupswe can prove an upper bound of 4 or less. Indeed we have Theorem D.
Let G be a finite simple group. Then ccn ( G ) ≤ G is a group ofLie type X r ( q ) where q ≤ f ( r ) for a suitable function f .Consequently, excluding these possible exceptions, we have w ( G ) · · · w ( G ) = G forall words w , . . . , w which are not an identity of any finite simple group.Treating the remaining groups seems very challenging. MICHAEL LARSEN, ANER SHALEV, AND PHAM HUU TIEP As ccn ( G ) reflects information on all simple subgroups of G , it cannot easily bedetermined from the character table of G , as iw ( G ) and similar invariants such as thecovering number and extended covering number of G can be. We were able to determineit in a few cases, of which the following results are representative. Proposition 1.4. If p ≥ is a prime, then ccn (PSL ( p )) = ( if p ≡ , if p ≡ . Proposition 1.5.
For all sufficiently large n , ccn ( A n ) = 3 . Proposition 1.6. If n ≥ and q − is sufficiently large and relatively prime to n ,then ccn (PSL n ( q )) = 4 . It would be interesting to find an example where ccn ( G ) > iw ( G ); at present, we donot know of any. 2. Preliminaries
Proposition 2.1.
Let G be a finite group. If ccn ( G ) < ∞ , then G is perfect. If G issimple, then ccn ( G ) < ∞ .Proof. For G finite, ccn ( G ) < ∞ if and only if S ( G ) generates G for every amplecharacteristic collection S .If G is not perfect, then by pulling back a subgroup of prime index from its abelian-ization, we obtain a normal subgroup G p ⊳ G of index p for some prime p . Then S ( H ) = { h p | h ∈ H } defines a characteristic collection of subgroups, and it is amplesince no finite simple group has exponent p . However, S ( G ) n is contained in G p for all n ∈ N , so ccn ( G ) = ∞ .If G is simple, let S be an ample characteristic collection. Then S ( G ) generates anon-trivial normal subgroup of G . As S is ample, this must be G itself. (cid:3) We recall that the extended covering number ecn ( G ) of a finite simple group G is thesmallest integer n such that C · · · C n = G if C , . . . , C n is any sequence of non-trivialconjugacy classes of G . Lemma 2.2. If G is a finite simple group, then ccn ( G ) ≤ ecn ( G ) − .Proof. If ccn ( G ) ≥ n , then there exist ample characteristic collections S i ( i = 1 , . . . , n −
1) such that S ( G ) · · · S n − ( G ) = G . As S , . . . , S n − are ample, S ( G ) · · · S n − ( G ) ⊇ C · · · C n − for some sequence C , . . . , C n − of non-trivial conjugacy classes of G . Moreover1 ∈ S ( G ) · · · S n − ( G ) . Therefore, there exists a non-trivial conjugacy class D such that D is disjoint from C · · · C n − . This implies 1 C · · · C n − D − , so ecn ( G ) ≥ n + 1. (cid:3) Lemma 2.3.
Let q ≥ be a prime power and C , C , C not necessarily distinct non-central conjugacy classes in SL ( q ) . HARACTERISTIC COVERING NUMBERS OF FINITE SIMPLE GROUPS 5 (i)
The product C C C contains all non-central elements of SL ( q ) . (ii) If q = 5 , then C C contains elements of order q − and order q + 1 . (iii) If q = 5 , then C C contains an element of order and one of order or order .Proof. Let G be a finite group. Recall that, by the extended Frobenius formula, thenumber of solutions to the equation g = x x · · · x k for a fixed g ∈ G and x i ∈ C i = c Gi (1 ≤ i ≤ k ) is given by(2.1) Q ki =1 | C i || G | · X χ ∈ Irr( G ) χ ( c ) · · · χ ( c k ) χ ( g − ) χ (1) k − . This implies that g ∈ C · · · C k provided the above expression is non-zero.The result now follows from the well known character table of G := SL ( q ). (cid:3) Lemma 2.4.
Let q ≥ be a prime power, d the g.c.d. of q − and , and C , C , C not necessarily distinct non-central conjugacy classes in G := PSL ( q ) . Let C q denotethe set of elements of order q . (i) The product C C C equals G . (ii) The product C C contains elements of order ( q − /d and ( q + 1) /d (iii) If q ≡ is prime, then C C contains an element of C q . (iv) If q ≡ is prime, then C C q contains G \ { } . (v) If q ≡ is prime and C , C are classes of involutions, then C C isdisjoint from C p .Proof. For q even, PSL ( q ) = SL ( q ), but in either case (i) and (ii) follow immediatelyfrom the corresponding statements in Lemma 2.3. For q an odd prime, C q is the unionof the two conjugacy classes. Conclusions (iii)–(v) follow by computing (2.1) in therelevant cases using the character table of G := PSL ( q ). (cid:3) From this lemma, we easily deduce Proposition 1.4.
Proof of Proposition 1.4.
The two conjugacy classes in C q are conjugate to one an-other under PGL ( q ), so if S ( G ) S ( G ) meets C q at all, it contains it. If S ( G ) and S ( G ) each contain a non-trivial conjugacy class of G , then by Gow’s theorem [G], S ( G ) S ( G ) contains all semisimple elements. If q ≡ S ( G ) S ( G ) = G , so ccn ( G ) ≤
2. On the other hand, ccn ( G ) ≥ iw ( G ) ≥
2, so ccn ( G ) = 2. If q ≡ ccn ( G ) ≤
3, and Lemma 2.4(v) implies iw ( G ) ≥
3, so ccn ( G ) = 3. (cid:3) Lemma 2.5. If q ≥ and C and C are non-trivial PGL ( q ) orbits in PSL ( q ) , then C C covers all split regular semisimple conjugacy classes in PSL ( q ) .Proof. Again, this follows from easy character table computations. (cid:3)
Lemma 2.6. (i) If C , C , C are non-trivial conjugacy classes in PSL ( q ) , then C C C contains all non-trivial elements of PSL ( q ) . (ii) If < q ≡ −
1( mod 3) and C , C , C are non-trivial conjugacy classes in PSU ( q ) ,then C C C contains all non-trivial elements of PSU ( q ) . MICHAEL LARSEN, ANER SHALEV, AND PHAM HUU TIEP (iii) If ∤ ( q + 1) and C , C , C are non-trivial conjugacy classes in SU ( q ) , then C C C contains all elements of order q − q +1 in SU ( q ) . Moreover, ecn (SU ( q )) =5 .Proof. Parts (i) and (ii) follow immediately from the fact [O, Corollary 1.9] that ecn (PSL ( q )) =4, and that ecn (PSU ( q )) = 4 for those q .(iii) In this case, GU ( q ) = SU ( q ) × Z (GU ( q )), and so each C i = g SU ( q ) i is a non-central GU ( q )-conjugacy class; furthermore, any element g of order q − q + 1 belongsto a class C ( k )8 in [O, Table 1]. By assumption, det( g i ) = 1 for 1 ≤ i ≤
3, and thuscondition (2) of [O, Theorem 1.3] holds. Choosing m = 4, we see that condition (3)of [O, Theorem 1.3] also holds; see the first remark on [O, p. 221]. Now [O, Theorem1.3] shows that 1 ∈ C C C · ( g − ) SU ( q ) , and so g ∈ C C C . The last statement iscontained in [O, Corollary 1.9]. (cid:3) Lemma 2.7.
Let G be a finite simple group of Lie type, and let s, t ∈ G be twosemisimple elements such that (i) For any pair of ample characteristic collections S , S , the conjugacy class of s belongs to S ( G ) S ( G ) , (ii) The product of the conjugacy class of s and the conjugacy class of t contains G r { } , (iii) The element s is regular.Then ccn ( G ) ≤ .Proof. By the theorem of Gow [G], every semisimple element in G , in particular t , can bewritten as a product of two conjugates of s . Now, if S , . . . , S are ample characteristiccollections, then s G ⊆ S S and s G ⊆ S S , whence t G lies in S ( G ) S ( G ) S ( G ) S ( G ).Therefore G r { } ⊂ s G · t G ⊆ S ( G ) · · · S ( G ) , and it follows that ccn ( G ) ≤ (cid:3) Lemma 2.8.
Let G be a finite group, and let d ( G ) the lowest degree of a non-trivialcharacter of G . Let q ≥ and ϕ : SL ( q ) → G be a non-trivial homomorphism. Supposethat (2.2) | C G ( ϕ ( s )) | / ≤ d ( G ) for any s ∈ SL ( q ) of order q − , or for any s ∈ SL ( q ) of order q + 1 if q = 5 , or forany s ∈ SL ( q ) of order or if q = 5 . Then ccn ( G ) ≤ .Proof. By Lemma 2.4, if S , S are ample characteristic collections, then g := ϕ ( s ) ∈ S ( G ) S ( G ), where s ∈ SL ( q ) can be chosen to have order q −
1, or q + 1 if q = 5, oreither 3 or 6 when q = 5. Let C := g G . By the Frobenius formula, C = G if X G = χ ∈ Irr( G ) | χ ( g ) | χ (1) < . HARACTERISTIC COVERING NUMBERS OF FINITE SIMPLE GROUPS 7
For any χ in the summation, | χ ( g ) | ≤ | C G ( g ) | / by the centralizer bound for charactervalues, and χ (1) ≥ d ( G ). Hence, by the second orthogonality relation, X G = χ ∈ Irr( G ) | χ ( g ) | χ (1) < | C G ( g ) | / d ( G ) · X χ ∈ Irr( G ) | χ ( g ) | = | C G ( g ) | / d ( G ) , and we can deduce the needed inequality from (2.2). (cid:3) We conclude this section with two results which allow us to prove that ccn ( G ) ≤ G is of Lie type and is sufficiently large compared to its rank. Proposition 2.9.
Let G denote a connected, simply connected simple algebraic groupof rank r over F q and G the finite simple group obtained by taking the quotient of G ( F q ) by its center. Let r = r + · · · + r k be a partition. If there exists a homomorphism φ : SL ( q r ) × · · · × SL ( q r k ) → G with central kernel and if q is sufficiently large interms of r , then ccn ( G ) ≤ .Proof. The number of F q -points on a torus of rank s is at most ( q + 1) s . The regularelements in a maximal torus T of G lie in a union of proper subtori of T indexed bypositive roots, so the number of elements in T ( F q ) which are not regular semisimple isless than 2 r ( q + 1) r − .We write H := Q ki =1 SL ( q r i ) /Z , where Z := ker φ is some subgroup of the productof the centers of the SL ( F q ri ) and is therefore of order at most 2 r . The inclusion of H in G gives a homomorphism k Y i =1 F × q ri → G. The image under this homomorphism of elements which are non-trivial in each coordi-nate is at least 2 − r (1 − q − ) r q r . However, the image lies in a maximal torus T of G , i.e., the image of the F q -points of amaximal torus T of G in G . Lemma 2.5 therefore gives a lower bound of the form c r q r forthe cardinality of the set of regular semisimple elements of T which lie in S ( H ) S ( H )where S and S are ample characteristic collections. The conjugacy class of each suchelement meets T in at most | W | elements, where W denotes the Weyl group of G withrespect to T , whose order is bounded in terms of r . Each conjugacy class of a regularsemisimple element in G ( F q ) has at least ( q + 1) − r | G ( F q ) | elements, so overall, we get apositive lower bound, depending only on r , for the proportion of elements of G which liein S ( G ) S ( G ). Since S ( G ) S ( G ) is a normal subset of G , [LST2, Theorem A] impliesthat S ( G ) S ( G ) S ( G ) S ( G ) contains all non-trivial elements of G . (cid:3) Note that the same argument works also for Suzuki and Ree groups.
Lemma 2.10.
Let G be a connected, simply connected simple algebraic group of rank r over F q and G the quotient of G ( F q ) by its center. Let φ : SL k → G be a homomorphismof algebraic groups over F q whose kernel lies in the center of SL k . Let G km denote a splitmaximal torus of SL k , and suppose φ ( G km ) is a maximal split torus of G which containsregular F q -points. Then if q is sufficiently large in terms of r , we have ccn ( G ) ≤ . MICHAEL LARSEN, ANER SHALEV, AND PHAM HUU TIEP
Note that if G is split, then φ ( G km ) is a maximal torus, so it contains regular elementsover F q if q is sufficiently large. If G is not split, a maximal split torus is not a maximaltorus, so k < r . Proof.
For q sufficiently large, there exists ( a , . . . , a k ) ∈ ( G m ( F q ) r { } ) k such that φ ( a , . . . , a k ) is regular. By Lemma 2.5, ( a , . . . , a k ) ∈ S (SL ( q ) k ) S (SL ( q ) k ) if S and S are ample characteristic collections. By a theorem of Ellers and Gordeev [EG,Theorem 1] the product of G -conjugacy classes of any two regular semisimple elementslying in maximal split tori of G covers G r { } . (cid:3) Classical groups
For every positive integer n , we denote by Φ n ( x ) the n th cyclotomic polynomial. Werecall [Zs] that for n > n, q ) = (6 , n ( q ) is always divisible by a Zsigmondyprime ℓ , meaning that ℓ does not divide q , which implies and the order of q (mod ℓ ) isexactly n . In particular, ℓ ≡ n ). Theorem 3.1. If G is a finite simple classical group, then ccn ( G ) ≤ .Proof. We consider all the six types A r , A r , B r , C r , D r , and D r . Let S , . . . , S beample characteristic collections and set d := gcd(2 , q − Case G = PSL n ( q ) with n ≥ n, q ) = (2 , , ecn (PSL ( q )) = 4for q ≥ ecn (PSL ( q )) = 4 [O, Corollary 1.9], so we may assume n ≥
4. For ( n, q ) = (6 , , (8) < G , and any element of order 7 lies in S ( G ) S ( G ) for S i ample characteristic collections. Using the character tables for thesetwo groups in the GAP library, we check that the square of this conjugacy class coversthe non-trivial elements of the group, so ccn ( G ) ≤ n, q ) = (6 , , q = p f for a prime p , and m := ⌊ n/ ⌋ so that m ≥ n ∈ { m, m + 1 } . By [Zs] and the as-sumption on ( n, q ), we can find a Zsigmondy prime ℓ for (2 mf, p ). By Lemma 2.4, S (PSL ( q m )) S (PSL ( q m )) contains an element of order ( q m + 1) /d . Since SL ( q m ) ≤ SL m ( q ) ≤ SL n ( q ), it follows that S ( G ) S ( G ) contains an element s of order divisibleby ( q m + 1) /d which is divisible by ℓ , and so s is regular semisimple. As discussed in[GT, § t ∈ G such that s G · t G ⊇ G r { } .Hence ccn ( G ) ≤ Case G = PSU n ( q ) with n ≥ n, q ) = (3 , ecn (PSU ( q )) ≤ n ≥
4. If G = PSU (2), then SL (8) < G , andwe easily check that the square of the class in G of any element of order 7 covers thenontrivial elements of G , so ccn ( G ) ≤ n, q ) = (6 , m := ⌊ n/ ⌋ , so that m ≥ n ∈ { m, m + 1 } . By [Zs] and the assumption on ( n, q ), we can find a Zsigmondyprime ℓ for (2 mf, p ) when 2 | m , and for ( mf, p ) when 2 ∤ m ; note that ℓ divides q m − Q m − i =1 (( − q ) i − S (PSL ( q m )) S (PSL ( q m )) contains anelement of order ( q m + ( − m ) /d . Note thatSL ( q m ) ≤ SL m ( q ) < SU m ( q ) ≤ SU n ( q ) HARACTERISTIC COVERING NUMBERS OF FINITE SIMPLE GROUPS 9 when 2 | m , and SL ( q m ) ∼ = SU ( q m ) ≤ SU m ( q ) ≤ SU n ( q )when 2 ∤ m . It follows that S ( G ) S ( G ) contains an element s of order divisible by( q m + ( − m ) /d which is divisible by ℓ , and so s is regular semisimple (as one can seeby checking the eigenvalues of s ). As discussed in [GT, § t ∈ G such that s G · t G ⊇ G r { } . Hence ccn ( G ) ≤ Case G = PSp n ( q ) with n ≥ n, q ) = (2 , (2) ′ ∼ = PSL (9)). ByLemma 2.4, S (PSL ( q n )) S (PSL ( q n )) contains an element s of order ( q n − /d and S (PSL ( q n )) S (PSL ( q n )) an element t of order ( q n + 1) /d . SinceSL ( q n ) ∼ = Sp ( q n ) < Sp n ( q ) ,S ( G ) S ( G ) contains regular semisimple elements s of order divisible by ( q n − /d , and S ( G ) S ( G ) contains a regular semisimple element of order divisible by ( q n + 1) /
2. Asdiscussed in [GT, § s G · t G ⊇ G r { } . Hence S ( G ) S ( G ) S ( G ) S ( G ) ⊇ G r { } ,and ccn ( G ) ≤ Case G = P Ω ǫn ( q ) with n ≥ ǫ = ± . First assume that n = 7 and 2 ∤ q . Note thatΩ ( q ) > SO +6 ( q ) > SL ( q ) , Ω ( q ) > SO − ( q ) > SU ( q ) . Hence, by Lemma 2.6, S ( G ) S ( G ) S ( G ) contains a regular semisimple element s oforder q + q + 1 and S ( G ) S ( G ) S ( G ) contains a regular semisimple element t of orderdivisible by ( q − q + 1) / gcd(3 , q + 1). As discussed in [GT, § s G · t G ⊇ G r { } .Hence ccn ( G ) ≤ n ≥
8, so that m := ⌊ n/ ⌋ ≥
2. Note thatPSL ( q m ) ∼ = Ω − ( q m ) ≤ Ω − m ( q ) , and S (PSL ( q m )) S (PSL ( q m )) contains an element ˜ s of order ( q m + 1) /d . Such anelement s is regular semisimple in each of the terms ofΩ − m ( q ) < Ω m +1 ( q ) < Ω ± m +2 ( q ) . Hence, if G ∈ { P Ω − m ( q ) , Ω m +1 ( q ) , P Ω ± m +2 ( q ) } , then S ( G ) S ( G ) contains a regularsemisimple element s of order ( q m + 1) /d . As discussed in [GT, §§ t ∈ G such that s G · t G ⊇ G r { } . Hence ccn ( G ) ≤ D r .Consider the case of G = P Ω +4 m ( q ), and note thatPSL ( q ) × PSL ( q m − ) ≤ Ω − ( q ) × Ω − m − ( q ) < Ω +4 m ( q ) . Hence, if m ≥
3, then applying Lemma 2.4 we see that S ( G ) S ( G ) contains an element s = ( x, y ), with x ∈ S (PSL ( q )) of order ( q + 1) /d and y ∈ S (PSL ( q m − )) of order( q m − + 1) /d . This element s is regular semisimple of type T − , − , m − , and, as shown in[LST1, Proposition 7.1.1] and [GM, Theorem 7.6], there exists a semisimple element t ∈ G such that s G · t G ⊇ G r { } . Hence ccn ( G ) ≤ m = 2, but q ≥
3. ThenSL ( q ) < SO +6 ( q ) ֒ → Ω +8 ( q ) , SU ( q ) < SO − ( q ) ֒ → Ω +8 ( q ) . By Lemma 2.6, S ( G ) S ( G ) S ( G ) contains an element s of order divisible by ( q + q + 1) / gcd(3 , q − T + , +3 , in G ,see [GT, § S ( G ) S ( G ) S ( G ) contains an element t of order divisible by( q − q + 1) / gcd(3 , q + 1), and such an element is regular semisimple of type T − , − , in G (since q ≥ s G · t G = G r { } , whence ccn ( G ) ≤
6. In thecase G = Ω +8 (2), one can check directly that ccn ( G ) ≤
4, using an element of order 7in SL (8) < G .Consider the case of G = Ω m +3 ( q ) with m ≥
2, and note thatPSL ( q m ) < Ω − m ( q ) × Ω ( q ) < Ω +4 m +3 ( q ) . Again using Lemma 2.4 we see that S ( G ) S ( G ) contains an element s ∈ S (PSL ( q m ))of order ( q m + 1) /d . Note that C SO m +3 ( q ) ( s ) = T × SO ( q ) , where T < SO − m ( q ) has order q m + 1. Since SO ( q ) ∩ G = Ω ( q ) has index 2 in SO ( q ),it follows that | C G ( s ) | = ( q m + 1)( q − q ) / . Let B := q m − . By [LOST, Corollary 5.8], Ω has exactly q + 4 nontrivial irreduciblecharacters of degree ≤ B , which are the characters D ◦ α with α ∈ Irr(Sp ( q )), listed in[LOST, Proposition 5.7]. The proof of Lemma 2.8 shows that(3.1) X χ ∈ Irr( G ) , χ (1) >B | χ ( s ) | χ (1) < | C G ( s ) | / B ≤ (cid:0) q m + 1)( q − q ) / (cid:1) / q m − < − / q m − / < . . The degrees of D ◦ α are listed in [LOST, Table I], showing that two of them have ℓ -defect0 for ℓ a Zsigmondy prime for ( n − , q ) and so vanish at s . Next we estimate | χ ( s ) | forthe remaining q + 2 characters. Note that, in the action of s on the natural G -module F nq , s has a unique eigenvalue λ that belongs to F q , and this eigenvalue is λ = 1 andhas multiplicity 3. Consider the action of x ⊗ s on V := F q ⊗ F nq for any x ∈ Sp ( q ) and F q being the natural module for Sp ( q ). Then the fixed point subspace of x ⊗ s on V has dimension 6 if x = 1, 3 if 1 = x is unipotent, and 0 otherwise. It follows from theformula [LOST, Lemma 5.5] for D α that(3.2) | D α ( s ) | ≤ α (1) q ( q − (cid:0) q + q / ( q −
1) + q ( q − − q (cid:1) = α (1)( q / + 2 q/ ( q + 1)) . Note that D ◦ α = D α , unless α (1) = ( q + 1) / D ◦ α = D α − G . Now, if q = 3, then α (1) ≤
3, and so (3.2) shows that | D ◦ α ( s ) | < .
5. Since D ◦ α (1) ≥ X χ ∈ Irr( G ) , <χ (1) ≤ B | χ ( s ) | χ (1) ≤ · (10 . < . . HARACTERISTIC COVERING NUMBERS OF FINITE SIMPLE GROUPS 11 If q ≥
5, then since α (1) ≤ q + 1, (3.2) shows that | D ◦ α ( s ) | ≤ q + q / ( q + 1). As D ◦ α (1) > q , it follows that X χ ∈ Irr( G ) , <χ (1) ≤ B | χ ( s ) | χ (1) ≤ ( q + 2) · (cid:0) q + q / ( q + 1) (cid:1) q < . . Together with (3.1), we have shown that X G = χ ∈ Irr( G ) | χ ( s ) | χ (1) < . , and so ( s G ) ⊇ G r { } by the Frobenius formula. Consequently, ccn ( G ) ≤ (cid:3) Proposition 3.2. If G is a classical group and the order of G is sufficiently large interms of the rank of G , then ccn ( G ) ≤ .Proof. We do a case analysis.
Case G = PSL n ( q ). Let m = ⌊ n/ ⌋ . The obvious homomorphism SL ( q ) m → SL n ( q )maps ( x , . . . , x m ) to a regular semisimple element in a split maximal torus of SL n over F q and hence to a split regular semisimple element of G , whenever the x i are regularsemisimple elements with eigenvalues λ ± i , λ i ∈ F × q , and λ i = λ ± j for all i, j . This casenow follows from Lemma 2.10. Case G = PSU n ( q ). Let m = ⌊ n/ ⌋ . We proceed exactly as before, using the obvioushomomorphism from SL ( q ) m = SU ( q ) m to SU n ( q ). An m -tuple of regular semisimpleelements such that λ i = λ ± j for all i, j maps to an element in an m -dimensional splittorus of G , and this is a maximal split torus [T, Table 2]. Case G = PSp n ( q ). We proceed as before, using the obvious homomorphism fromSL ( q ) n ∼ = Sp ( q ) n to Sp n ( q ), which maps any split maximal torus of SL n to a splitmaximal torus of Sp n . Case G = P Ω n +1 ( q ). If n = 2 m , we proceed as before, using composition of theobvious homomorphisms from SL ( q ) n toSpin ( q ) ∗ Spin +4 ( q ) ∗ . . . ∗ Spin +4 ( q ) | {z } m − ∼ = Sp ( q ) ∗ SL ( q ) ∗ . . . ∗ SL ( q ) | {z } n − and Spin ( q ) ∗ Spin +4 ( q ) ∗ . . . ∗ Spin +4 ( q ) | {z } m − → Spin n +1 ( q ). If n = 2 m + 1, we map SL ( q ) n onto Spin ( q ) ∗ Spin +4 ( q ) ∗ . . . ∗ Spin +4 ( q ) | {z } m and maps the latter to Spin n +1 ( q ). Case G = P Ω +2 n ( q ). If n = 2 m , we map SL ( q ) n onto Spin +4 ( q ) ∗ . . . ∗ Spin +4 ( q ) | {z } m andmap the latter to Spin +2 n ( q ). If n = 2 m + 1, we map SL ( q ) n − toSL ( q ) ∗ SL ( q ) ∗ . . . ∗ SL ( q ) | {z } n − ∼ = Spin +6 ( q ) ∗ Spin +4 ( q ) ∗ . . . ∗ Spin +4 ( q ) | {z } m − , which embeds in the usual way in Spin +2 n ( q ). Case G = P Ω − n ( q ). If n = 2 m + 1, we map SL ( q ) n − = SU ( q ) × SL ( q ) n − ontoSU ( q ) ∗ SL ( q ) ∗ . . . ∗ SL ( q ) | {z } n − ∼ = Spin − ( q ) ∗ Spin +4 ( q ) ∗ . . . ∗ Spin +4 ( q ) | {z } m − , which embeds in the usual way in Spin − n ( q ). Note that an ordered pair of regularsemisimple elements of SU ( q ) with distinct eigenvalue pairs gives a regular semisimpleelement of SU ( q ), and it follows that the image of a maximal torus of SL n − in Spin − n ( q )meets the regular semisimple locus. As Spin − n is not split, it cannot have a split torusof rank n , so any split torus of rank n − n − is a maximal split torus of Spin − n .If n = 2 m , we identify Spin − ( q ) with SL ( q ) and map SL ( q ) × SL ( q ) n − intoSpin − n ( q ). This maps ( y, x , . . . , x n − ) to a regular semisimple element when y and the x i are regular semisimple, the eigenvalues of all x i lie in F q , the eigenvalues of y donot lie in F q , and no two x i have an eigenvalue in common. We can no longer use theEllers-Gordeev method, but Proposition 2.9 applies, so we still obtain ccn ( G ) ≤ (cid:3) Exceptional groups of Lie type
The following lemma will be useful for us.
Lemma 4.1.
Let G be an exceptional group of Lie type and s an element of G whoseorder is divisible by a prime ℓ which is Zsigmondy for ( n, q ) . Then (i) If G = F ( q ) and n = 8 , then s is regular semisimple with centralizer order q + 1 . (ii) If G = F ( q ) with q ≥ , n = 4 , and ℓ > , then s is regular semisimple withcentralizer order q + 1 . (iii) If G = E ( q ) or G = E ( q ) , and n = 8 , then s is regular semisimple withcentralizer order ( q − q + 1) . (iv) If G = E ( q ) and n = 7 , then s is regular semisimple with centralizer order q − . (v) If G = E ( q ) and n = 7 , then s has centralizer order dividing ( q − q )( q − .Proof. In all five cases, ℓ − ≥
5, so ℓ ≥
11. Let t denote a power of s ofexact order ℓ , so t is semisimple. By [MT, Lemma 2.2], the connected center T of thecentralizer of t has order divisible by ℓ . It suffices to prove the stated claims for t , sincethe centralizer of t contains the centralizer of s . In general, ℓ divides Φ m ( q ) only if m is n times a power of ℓ . The order of every torus T in G can be written Q j Φ i j ( q ), where P φ ( i j ) is the rank of the torus; in particular, φ ( i j ) ≤
8, which implies that i j < nℓ , soif ℓ divides | T , then i j = n for some i j .In the F case, φ (8) = 4 so T must have order Φ ( t ), so t must be regular semisimple.The same argument applies to F ( q ) (which is F ( r ) with r = q / ). In the E and E cases, we consult the connected centralizers in the E table in [FJ1] and concludethat t must be regular semisimple and associated to w , implying the stated centralizerorder. The E -table in [FJ1] has a Φ ( q ) factor in the connected centralizer of t onlyfor the regular semisimple case associated to w . The E -table in [FJ2] has a Φ ( q )factor in three cases: the regular semisimple classes associated to w and w , with HARACTERISTIC COVERING NUMBERS OF FINITE SIMPLE GROUPS 13 centralizer orders ( q − q −
1) and ( q + 1)( q −
1) respectively and the ( A , w ) class,with centralizer order ( q − q )( q − (cid:3) Theorem 4.2. If G is a finite simple group of exceptional Lie type, then ccn ( G ) ≤ .Proof. We consider each of the ten possibilities.
Case B ( q ). As G is a finite simple group, we have q ≥
8. By [AH, p. 2], ecn ( G ) = 4,so by Lemma 2.2, ccn ( G ) ≤ Case G ( q ) ′ . The character table for q = 2 is in the GAP library, and we easily computethat the extended covering number of this group is 5, so ccn ( G ) ≤
4. Otherwise, q ≥ G ( q ) ′ = G ( q ). By [LSS, Table 5.1], we have SL ( q ) < G ( q ). Let e := gcd(3 , q −
1) =gcd(3 , q + q + 1). By Lemma 2.6, for all S , S , S , every element s of order q + q +1 e inPSL ( q ) belongs to S (PSL ( q )) S (PSL ( q )) S (PSL ( q )), and let s be any lift of s toSL ( q ). By [GM, Table 10], every non-trivial element of G is a product of two suchelements s , so ccn ( G ) ≤ Case G ( q ) ′ . When q = 3, this is PSL (8), so we have ccn ( G ) ≤
3. When q ≥ G ( q ) is already simple. By [LSS, Table 5.1], PSL ( q ) < G , so by Lemma 2.4, any S ( G ) S ( G ) contains an element s of order q +12 , and by [GM, Theorem 7.1], everyelement of G r { } is a product of two conjugates of s . Thus ccn ( G ) ≤ Case D ( q ). We compute ecn ( D (2)) = 7 , ecn ( D (3)) = 6 , using the character table in the GAP library for the former and Frank L¨ubeck’s char-acter table, computed using the generic character table for D in CHEVIE [GHLMP],for the latter. So we may assume q ≥ ( q ) × SL ( q ) < D ( q ) if q is even and the centralproduct SL ( q ) ∗ SL ( q ) if q is odd. In either case, by Lemma 2.3, given ample char-acteristic collections S and S , this subgroup contains an element s of S ( G ) S ( G ) oforder ( q − q + 1) /d where d := gcd(2 , q − s in the Deriziotis-Michler classification [DM, Table 2.1]; it follows that the orderof its centralizer is ( q − q + 1) < q . From L¨ubeck’s table of degrees for D ( q ) [Lu],we know that except for the trivial character and a character χ of degree q Φ ( q ), allother irreducible characters of D ( q ) have degree greater than q / q ≥
4. By [Sp,Table 2], the value of χ at a regular semisimple element with centralizer ( q − q + 1)is 1. Certainly, P χ ∈ Irr( G ) | χ ( s ) | = | C G ( s ) | < q , and | χ ( s ) | < (( q − q + 1)) / < q for all 1 G = χ ∈ Irr( G ). We conclude that X χ =1 | χ ( s ) | χ (1) < q Φ ( q ) + X χ (1) >ǫ (1) | χ ( s ) | χ (1) < q + q q / X χ | χ ( s ) | < q + 2 q < , so the Frobenius formula implies that ( s G ) = G . Case F ( q ). By [LSS, Table 5.1], Sp ( q ) < F ( q ). Therefore, SL ( q ) < F ( q ). We aretherefore guaranteed elements s of order q + 1 in S ( F ( q )) S ( F ( q )). By Lemma 4.1,such elements are regular semisimple. By [GM, Table 11], If t is any element of orderΦ ( t ), then the product of the conjugacy class of s and the conjugacy class of t covers F ( q ) r { } . Thus, Lemma 2.7 implies ccn ( F ( q )) ≤ Case F ( q ) ′ . For q = 2, this is the Tits group. In this case, we can use the charactertable in GAP to compute the extended covering number, which is 5, so ccn ( G ) ≤ q ≥
8, so F ( q ) is already perfect. By [LSS, Table 5.1],SL ( q ) < Sp ( q ) < F ( q ). Thus, F ( q ) contains regular semisimple elements s oforder q + 1 by Lemma 4.1. The centralizer of s in F ( q ) has order q + 1, and by [LS]we have d ( G ) ≥ − / (1 − q − ) q / . Lemma 2.8 now implies that ccn ( G ) ≤ q ≥ Cases E ( q ) and E ( q ). As F ( q ) < G , we have SL ( q ) < G , and proceeding as in the F ( q ) case, we are guaranteed elements s of order q + 1 in S ( G ) S ( G ). These elementsare regular semisimple by Lemma 4.1. By [GM, Table 11], there exists a semisimpleelement t such that the product of the conjugacy classes of s and t cover G r { } . Thus, ccn ( G ) ≤ Case E ( q ). By [LSS, Table 5.1], E ( q ) contains PSL ( q ). If S and S are amplecharacteristic collections, then S (PSL ( q )) S (PSL ( q )) contains an element s of order q − q − . By Lemma 4.1, this element is regular semisimple. In [GM, Table 11] isit shown that there exists an order x prime to q such that the product of any conjugacyclass of order x = q − x covers G r { } . Theproof depends only on the divisibility of x by a Zsigmondy prime and therefore goesthrough unchanged if x = q − . Lemma 2.7 now implies that ccn ( E ( q )) ≤ Case E ( q ). By [LSS, Table 5.1], E ( q ) contains a E sc7 ( q ), which, in turn, containsa perfect central extension ˜ H of H = PSL ( q ). We can therefore regard ˜ H as acentral quotient of SL ( q ). If S and S are ample characteristic collections, then S ( ˜ H ) S ( ˜ H ) contains an element s of order q − q − . This element is thereforesemisimple, and by Lemma 4.1, the order of its centralizer divides ( q − q )( q − d ( G ) ≥ q ( q − ≥ q . As ( q ) / = q < q , Lemma 2.8 implies ccn ( E ( q )) ≤ (cid:3) In most cases, we can prove the improved bound of 4.
Proposition 4.3. If G is a sufficiently large finite simple group of exceptional Lie type,then ccn ( G ) ≤ .Proof. For each series of exceptional groups of Lie type with the exception of B (whichis already covered by Theorem 4.2), E and E , [LSS, Table 5.1] gives a subgroup ofthe type for which Proposition 2.9 applies.Suppose G is of type E , i.e., it is the quotient of G ( F q ) by its center, where G is thesplit simply connected group group over F q of type E . As A × A can be obtained fromthe extended Dynkin diagram of E by deleting a vertex, it follows that there exists ahomomorphism SL × SL → G of algebraic groups over F q , and a maximal torus of HARACTERISTIC COVERING NUMBERS OF FINITE SIMPLE GROUPS 15 the former maps to a maximal torus of the later. Thus, there exists a homomorphism φ : SL → G which factors through SL × SL → G , which maps a split maximal torusinto a split maximal torus, and whose image contains regular elements if q is sufficientlylarge. The proposition now follows from Lemma 2.10.Finally suppose G is of type E . Let G be a simply connected non-split group oftype E over F q . By [T, Table 2], it is of rank 4 over F q . The rank of SL × SU over F q is also 4, so there is a homomorphism SL × SU → G mapping a maximal split torus toa maximally split maximal torus. Using SL = SU → SU , we obtain a morphism ofalgebraic groups over F q from SL to G which sends any split maximal torus of SL toa maximal split torus of G and whose image in G contains regular semisimple elements.The proposition again follows from Lemma 2.10. (cid:3) From this, we easily deduce Proposition 1.6.
Proof of Proposition 1.6.
Because n and q − n ( q ) = SL n ( q ),so 4 = iw (SL n ( q )) = iw (PSL n ( q )) ≤ ccn (PSL n ( q )) ≤ . (cid:3) Alternating groups and sporadic groups
Proposition 5.1.
For n ≥ we have ccn ( A n ) ≤ .Proof. By [AH, p. 1], we have ecn ( A n ) ≤ n ≤
9, so by Lemma 2.2, we may assume n ≥ p is a prime, the permutation representation on P ( F p ) embeds PSL ( p ) in A p +1 .Every element of order p maps to a p -cycle in A p +1 and therefore to a p -cycle in A n forall n ≥ p + 1. By Lemma 2.4, if p ≡ s ∈ S (PSL ( p )) S (PSL ( p )) , t ∈ S (PSL ( p )) S (PSL ( p ))of order p , so every element in A n which is a product of two p -cycles lies in S ( A n ) · · · S ( A n ).By a theorem of Bertrand [B], if ⌊ n ⌋ ≤ p ≤ n , then every element of A n is a productof two p -cycles. Therefore, ecn ( A n ) ≤ ⌊ n/ ⌋ , n − p = 13 and p = 17, we get the desired inequalityfor n in { , , , } and { , , , , , } respectively. The following table givesprimes which together cover all values of n ∈ [30 , . · ].
29 37 41 53 6173 97 113 149 197257 337 449 593 7731021 1361 1801 2393 31814241 5653 7537 10037 1338117837 23773 31657 42209 5626975017 99989 133277 177677 236897315857 421133 561461 748613 9981171330789 1774373 2365829 3154433 42059095607853 7477121 9969457 13292593 1772344923631253 31508329 42011093 56014789 7468635799581809 132775693 177034217 236045497 314727293419636389 559515161 746020213 994693597By a theorem of Ramar´e and Rumely [RR], if n ≥ , then the sum of log p overall p < n which are 1 (mod 4) lies between . n and . n . It follows that there isat least one such prime in the interval [ n, . n ] for all n > , and this is enough toshow ccn ( A n ) ≤ n > .If q is any odd prime power, then for C and C non-trivial conjugacy classes inPSL ( q ), there is an element of order q +12 in C C . This maps into a product of twodisjoint q +12 cycles. A theorem of Xu [X] asserts that if n − ≤ r + s ≤ n , then everyelement of A n is a product of two elements, each consisting of two disjoint cycles oflength r and s . Applying this for q = 9, q = 11, q = 23, q = 25, and q = 27 we get ccn ( A n ) ≤ n in { , } , { , } , { , } , { , } , and { , } respectively, soall cases are covered. (cid:3) To deal with A n for large n , it is useful to have the following lemma: Lemma 5.2.
Let p be a prime, k and n positive integers, n ≥ kp , and u , . . . , u k , v , . . . , v k ∈{ , . . . , n } such that u i = u j , v i = v j , and u i = v j when i = j . Then there exists anelement σ ∈ S n such that σ ( u i ) = v i for ≤ i ≤ k and σ p is the identity.Proof. For each i such that u i = v i , let X i = { u i } , and for each i such that u i = v i , let X i be a p -element set containing u i and v i and such that the X i are disjoint from oneanother. We define σ to be the identity on { , . . . , n } \ S X i and to cyclically permutethe elements of each X i such that σ ( u i ) = v i . (cid:3) We can now prove Proposition 1.5.
Proof of Proposition 1.5.
First we note by [TZ, Theorem 1.2] that iw ( A n ) ≥ n ≥
15: indeed, not every element in A n with n ≥
15 is a product of two involutions.Hence it suffices to show that ccn ( A n ) ≤ n is large enough.We say a characteristic collection S if of type r if r is the smallest prime such that S ( A ) contains an element of order r . If S is ample it must be of type 2, type 3, or type5. For any positive integer n and 0 ≤ m ≤ ⌊ n/ ⌋ , the embedding A m < A n guaranteesthat S ( A n ) contains an element of cycle type 1 − m m if S is of type 2, 1 − m m if S is of type 3, and 1 − m m , if S is of type 5. HARACTERISTIC COVERING NUMBERS OF FINITE SIMPLE GROUPS 17
For p ≡ ± A < PSL ( p ) < A p +1 , and every element of A of order r , when regarded as an element x of A p +1 , has at most 2 fixed points and otherwiseconsists entirely of r -cycles. For p = 59, x is therefore a permutation of the form r /r .If S is of type 2, the embedding A i × A n − i < A n gives S ( A n ) elements of cycle type1 − m m whenever m belongs to I i = (cid:20) i, i + (cid:4) n − i (cid:5)(cid:21) . When 75 + 60 k ≤ n <
75 + 60( k + 1), the consecutive intervals in the sequence I , . . . , I k overlap, so for 0 ≤ m ≤ k , all elements of cycle type 1 − m m belong to S ( A n ). Thismeans that every even permutation of order 2 with at least B = 75 fixed points lies in S ( A n ). By the same reasoning, for B = 100 and B = 60, all elements of order 3 and 5in A n with at least B and B fixed points respectively lie in S ( A n ) if S is respectivelyof type 3 or 5.By [LS1, Theorem 1.2(ii)], if x is an element of S n of prime order r and a boundednumber of fixed points, then for every irreducible character χ of S n , | χ ( x ) | ≤ χ (1) /r + o (1) .Therefore, if n is sufficiently large and x , x , x are three such elements in A n , of orders r , r , r respectively, and 1 r + 1 r + 1 r < , then the product of the S n -conjugacy classes of the x i cover A n r { } . (Indeed, notethat we have 1 /r + 1 /r + 1 /r ≤ /
42 in such a case, and so, by [LiSh2, Theorem2.6], X χ ∈ Irr( S n ) , χ (1) > | χ ( x ) χ ( x ) χ ( x ) | χ (1) < X χ ∈ Irr( S n ) , χ (1) > χ (1) / → n → ∞ .) We may assume without loss of generality that r ≤ r ≤ r , so weneed only consider the cases ( r , r ) = (2 , r , r ) = (2 , r = r = r = 3. Weproceed by case analysis. Case (2 , , r ). For any k ≥
1, let σ k = (1 , · · · (2 k − , k ) ∈ S k , τ k = (2 , · · · (2 k, k + 1) ∈ S k +1 . Thus, σ k τ k is a (2 k + 1)-cycle, while σ k τ k − is a 2 k -cycle. Applying this to any cycle, wesee that every element in S n can be written as a product of two involutions. Regarding S n − as the stabilizer in A n of { n − , n } , we see that every element of A n which fixes n − n can be written as a product of two involutions in A n . More generally, everyelement of A n with at least 2 + B fixed points can be written as a product of eveninvolutions, each of which has at least B fixed points and hence belongs to S ( A n ).We claim that if n is sufficiently large, for every ρ ∈ A n and p ∈ { , , } there exists π ∈ A n of order dividing p with at least B p fixed points such that πρ has at least 2 + B fixed points. We choose a sequence x , . . . , x B of distinct elements of { , . . . , n } such that ρ ( x i ) = x j for i = j and then a sequence y , . . . , y B p of distinct elements of { , . . . , n } r { ρ ( x ) , . . . , ρ ( x B ) } . By Lemma 5.2, if n is sufficiently large, there existsan even permutation π of order 1 or p which fixes each y i and maps each ρ ( x j ) to x j . Bythe above discussion, πρ ∈ S ( A n ) S ( A n ) and π − ∈ S ( A n ), hence ρ ∈ S ( A n ) S ( A n ) S ( A n ). Case (2 , , r ). Let σ k = (1 , , · · · (3 k − , k − ,σ ′ k = (3 , , · · · (3 k, k + 1) ,τ k = (2 , , , , · · · (3 k − , k, k + 1) ,τ ′ k = (1 , , , , · · · (3 k − , k − , k ) . Then σ k τ k is a (3 k + 1)-cycle, σ k +1 τ k is a (3 k + 2)-cycle, and σ ′ k τ ′ k +1 is a (3 k + 3)-cycle.Thus, every element of S n is a product of an involution σ and an element τ of order 1or 3. Since any such τ lies in A n , the same statement holds in A n , and if the producthas at least max( B , B ) fixed points, the same can be assumed of σ and τ . ApplyingLemma 5.2 as before, we can guarantee for each ρ ∈ A n the existence of an element π ∈ A n of order dividing r with at least B r fixed points such that πρ has at leastmax( B , B ) fixed points, and then finish as in the previous case. Case (3 , , A n can be written as a productof two elements of order dividing 3. Let σ k,l = (1 , , , , · · · (4 k − , k − , k − k − , k − , k − k, k + 1 , k + 2)(4 k + 4 , k + 5 , k + 6) · · · (4 l, l + 1 , l + 2) ,τ k,l = (3 , , , , · · · (4 k − , k − , k − k − , k, k + 1)(4 k + 2 , k + 3 , k + 4)(4 k + 6 , k + 7 , k + 8) · · · (4 l + 2 , l + 3 , l + 4) , where the first line in each expression is omitted if k = 0 and the second line is omittedif l = k −
1. Then σ k,k − τ k − ,k − is a (4 k − σ k,k − τ k,k − is a (4 k + 1)-cycle,so all odd cycles can be written as a product of two elements of order dividing 3. For l ≥ k ≥ k + l ≥ σ k,l τ k,l is a disjoint product of a 4 k -cycle and a (4 l + 4 − k )-cycle; σ k,l τ k − ,l is a disjoint product of a (4 k − l +6 − k )-cycle; σ k,l τ k,l − is a disjoint product of a 4 k -cycle and a (4 l + 2 − k )-cycle; and σ k,l τ k − ,l − is a disjointproduct of a (4 k − l + 4 − k )-cycle. Thus, all possible permutationsthat are products of two disjoint even-length cycles can be written as a product of twopermutations of order dividing 3. (cid:3) Proposition 5.3.
For all sporadic finite simple groups we have ccn ( G ) ≤ .Proof. By a theorem of Zisser [Zi], we have ecn ( G ) ≤ and Fi , so it suffices to consider these two cases. It is known [ATLAS, pp. 74, 163]that there are inclusions PSL (25) < F (2) ′ < Fi . By Lemma 2.4, if S and S are ample characteristic collections, then S (PSL (25)) S (PSL (25)) has an element oforder 13, and a machine computation shows that the product of any two conjugacyclasses of elements of order 13 in Fi contains Fi r { } . It is also known [ATLAS,p. 177] that PSL (17) < Fi . By Lemma 2.4, the product S (PSL (17)) S (PSL (17))contains an element of order 17; a machine computation shows that the square of theunique conjugacy class of order 17 in Fi is the whole group. (cid:3) Together with Theorems 3.1 and 4.2, Propositions 5.1 and 5.3 complete the proof ofTheorem A. Theorem D follows from results 5.1, 5.3, 3.2 and 4.3.
HARACTERISTIC COVERING NUMBERS OF FINITE SIMPLE GROUPS 19
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Email address : [email protected] Einstein Institute of Mathematics, Hebrew University, Givat Ram, Jerusalem 91904,Israel
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