An upper bound for the nonsolvable length of a finite group in terms of its shortest law
aa r X i v : . [ m a t h . G R ] F e b AN UPPER BOUND FOR THE NONSOLVABLE LENGTH OFA FINITE GROUP IN TERMS OF ITS SHORTEST LAW
FRANCESCO FUMAGALLI, FELIX LEINEN, AND ORAZIO PUGLISI
In memory of our friend Carlo Casolo
Abstract.
Every finite group G has a normal series each of whose factors iseither a solvable group or a direct product of non-abelian simple groups. Theminimum number of nonsolvable factors, attained on all possible such series in G , is called the nonsolvable length λ ( G ) of G . In the present paper, we provea theorem about permutation representations of groups of fixed nonsolvablelength. As a consequence, we show that in a finite group of nonsolvable lengthat least n , no non-trivial word of length at most n (in any number of variables)can be a law. This result is then used to give a bound on λ ( G ) in terms of thelength of the shortest law of G , thus confirming a conjecture of Larsen (see[2]).MSC2020: 20F22, 20B05, 20D99.Keywords: nonsolvable length, group laws. Introduction
Let F ∞ be the free group of countably infinite rank, with free generators x i ( i ∈ N ). Consider a word w = w ( x , . . . , x k ) ∈ F ∞ . For any group G , the map φ w : G k −→ G , defined by φ w ( g , . . . , g k ) = w ( g , . . . , g k ), is called the word map induced by w . The verbal subgroup w ( G ) of G is generated by the image of φ w .The word w is said to be a law in G , when w ( G ) = 1.In recent years, quite a large number of papers was devoted to questions related toword maps, leading to an impressive series of deep results. Particularly remarkableachievements were the proof of Ore ’s conjecture (see [13]), that every element in afinite non-abelian simple group is a commutator, and a series of results on
Waring type problems (see [10] and [11]). The interested reader should consult the survey[15] by
Shalev and its extensive bibliography in order to get a clear picture of theresearch in this area.Given a word w and a group G , one may ask under which circumstances w is nota law in G : Which properties of G ensure, that w ( G ) = 1? One of the first resultsin this direction is contained in [8]: Given a non-trivial word w , there exist only afinite number of finite non-abelian simple groups, which admit w as a law. In otherwords, for every non-trivial w , there exists a natural number N = N ( w ) such that Date : February 23, 2021. fumagalli | leinen | puglisi the verbal subgroup w ( S ) of any finite non-abelian simple group S of order largerthan N is non-trivial.Properties of word maps in arbitrary finite groups have been investigated in [2]:In particular, for certain words w = w ( x , . . . , x k ) ∈ F ∞ , the nonsolvable length λ ( G ) of G (see above abstract) can be bounded in terms of the sizes of the fibersof the verbal map φ w : G k −→ G . This provides evidence in favor of the followingconjecture of Michael Larsen (see [2, Conjecture 1.3]).
Conjecture.
There exists a function g : N −→ N such that λ ( G ) ≤ g ( ν ( G )) forevery finite group G . Here, ν ( G ) = min (cid:8) | w | (cid:12)(cid:12) w is a non-trival reduced law in G (cid:9) ,where | w | denotes the length of the reduced word w ∈ F ∞ . In the present paper, we shall confirm this conjecture in the following form.
Theorem A.
Let G be any finite group. Then λ ( G ) < ν ( G ) . Our bound on the non-solvable length can probably be improved: At least when w is not a commutator word, we can use [6, Proposition 5.10] in order to see, thatthe nonsolvable length λ ( G ) is bounded by log ( | w | ). This argument uses the fact,that a law of the form x n ˜ w with ˜ w ∈ F ′∞ always implies the law x n .This observation encourages us to set up the following conjecture. Conjecture.
The nonsolvable length of any finite group G can be bounded by afunction in O (log( ν ( G ))) . We shall obtain Theorem A as a consequence of the following main result.
Theorem B.
Let G be a finite group of nonsolvable length λ ( G ) = n . Then, forevery non-trivial word w = w ( x , . . . , x k ) ∈ F ∞ of length at most n , there exists g = ( g , . . . , g k ) ∈ G k such that w ( g ) = w ( g , . . . , g k ) = 1 . When proving Theorem B, one realizes immediately, that a detailed knowledgeof the action of the group on the components of certain non-abelian compositionfactors is needed. For this reason, we shall derive Theorem B from a result aboutpermutation representations of groups of fixed nonsolvable length, which might beof interest in its own right.In order to formulate this result, we introduce some further notation. Let w = y y · · · y n (cid:0) with y i ∈ { x ± , . . . , x ± k } (cid:1) be a reduced word in F ∞ . Then we shall needto consider its partial subwords , namely the words w i = y y . . . y i for 0 ≤ i ≤ n (where w = 1). Definition 1.
Let G be a finite group acting faithfully on the set Ω .We say that G satisfies property P n in its action on Ω (cid:0) and write G ∈ P n (Ω) (cid:1) ,if for every non-trival reduced word w = w ( x , . . . , x k ) ∈ F ∞ of length n , there exist ω ∈ Ω and g ∈ G k , such that the sequence { ωw i ( g ) } ni =0 consists of n + 1 distinctelements in Ω .We write G ∈ P n whenever G ∈ P n (Ω) for every faithful G -set Ω . n upper bound for the nonsolvable length Clearly, if G ∈ P n (Ω) for some faithful G -set Ω, then w ( G ) = 1 for every non-trivial reduced word w of length at most n . However, this property is much strongerthan the non-triviality of w ( G ). Indeed, when G ∈ P n (Ω), it is possible to find some g ∈ G k such that all the elements w ( g ) , w ( g ) , . . . , w n ( g ) = w ( g ) are non-trivialand pairwise distinct.Theorem B will be a consequence of the following result. Theorem C.
Let G be a finite group with λ ( G ) = n , and let Ω be a faithfultransitive G -set. Then, for every ω ∈ Ω and for every non-trivial reduced word w = w ( x , . . . , x k ) ∈ F ∞ of length n , there exist a Sylow -subgroup P of G anda tuple g ∈ P k such that the points ωw ( g ) , ωw ( g ) , . . . , ωw n ( g ) are pairwisedistinct. In particular, G ∈ P n . This theorem says: If G has nonsolvable length n , and if H is a core-free subgroupof G , then for every non-trivial word w = w ( x , . . . , x k ) of length n there exists a k -tuple g of elements from G such that the cosets H, Hw ( g ) , . . . . . . , Hw n − ( g ) , Hw n ( g ) = Hw ( g )are pairwise distinct.The proof of Theorem C relies heavily on the existence of so called rarefied subgroups. In [6, Theorem 1.1] it has been proved, that every finite group ofnonsolvable length m contains m -rarefied subgroups. These are subgroups of thesame nonsolvable length m , with a very restricted structure. In particular, thenon-abelian composition factors of an m -rarefied group belong to the class L = (cid:8) L (2 r ) , L (3 r ) , L ( p a ) , L (3) , B (2 r ) (cid:12)(cid:12) p, r odd primes , a ∈ N (cid:9) . The precise definition of m -rarefied groups will be given in section 3.In proving Theorem C, we shall reduce ourselves to the case of rarefied groupsand then use the constraints on the structure of such groups in order to completethe proof. The proof of Theorem C depends on the classification of finite simplegroups.We finally note, that Theorem C partly solves a question of Evgenii Khukhro and
Pavel Shumyatsky . They have shown in [9, Theorem 1.1], that the nonsolv-able length of any finite group G is bounded by 2 L ( G ) + 1, where L ( G ) denotesthe maximum of the 2-lengths of the solvable subgroups of G . This bound was im-proved in [6, Theorem 5.2] to L ( G ). In this context, Khukhro and
Shumyatsky raised the following question [9, Problem 1.3].
Problem.
For a given prime p and a given proper group varity V , is there a boundfor the non- p -solvable length of finite groups whose Sylow p -subgroups belong to V ? Theorem C gives an affirmative answer in the case when p = 2. fumagalli | leinen | puglisi Preliminaries on group actions and words
Consider any non-trivial reduced word w = w ( x , . . . , x k ) ∈ F ∞ of length n . Let w = y · · · y n with y j = x ǫ j i j and ǫ j ∈ {± } for all j . As before, we let w = 1 and w j = y · · · y j and e w j = ( w − j if ǫ j = − w − j − if ǫ j = +1 for 1 ≤ j ≤ n. For k -tuples g = ( g , . . . , g k ) and h = ( h , . . . , h k ) of elements in a group G , welet gh = ( g h , . . . , g k h k ).The proof of the following lemma is straightforward (see [12, Lemma 2.1]). Lemma 1.
Let N E G . Consider g ∈ G k and b ∈ N k . Then, in the above notation, w ( bg ) = a ǫ · · · a ǫ n n w ( g ) ∈ N w ( g ) with a j = ( b i j ) e w j ( g ) for all j. The following situation will come up several times in the course of our analysis:The normal subgroup of a finite group G is a direct product of subgroups M i (0 ≤ i ≤ m ), which are all isomorphic to a certain group M and which are permutedtransitively under the conjugation with elements from G . Lemma 2.
In the above situation, let H = N G ( M ) and c M = Q i =0 M i . Suppose,that F is a field and W is an irreducible F H -module satisfying [ W, M ] = 0 and [ W, c M ] = 0 . Then the induced F G -module V = Ind GH ( W ) is irreducible.Proof. Let T be a transversal to H in G containing 1. Then V = Ind GH ( W ) = W ⊗ H F G = ⊕ t ∈ T ( W ⊗ H t )and each of the subspaces W t = W ⊗ H t ( t ∈ T ) is a G -block. If i ∈ { , , . . . , m } and t ∈ T , then [ W t , M i ] = 0 whenever M i = M t . The subgroup M has trivialcentralizer in W , because W is an irreducible H -module, M is normal in H and acts non-trivially on W . If U is a G -submodule of V and u ∈ U is a non-trivial element, write u = P t ∈ T u t , with u t ∈ W t for all t ∈ T . Without loss ofgenerality, we may assume that u = 0 and pick x ∈ M not centralizing u . Then[ u, x ] = u ( x −
1) is still in U , hence U ∩ W = 0. Since U ∩ W is an H -module,it must be the whole W . Once we know that W ≤ U we immediately deduce U = V . (cid:3) The following lemma generalizes an idea from [1].
Lemma 3.
Let
G, H, W and V be as in Lemma 2 and adopt the notation introducedin its proof. In particular, T denotes a transversal to H in G with ∈ T . Suppose,that the subgroup A of M satisfies [ W, A ] = 0 . Let M ∗ = Y t ∈ T M t and A ∗ = Y t ∈ T A t . Suppose further, that w = y · · · y n is a non-trivial reduced k -variable word of length n , and that there exists some g ∈ ( N G ( A ∗ )) k such that the groups M w j ( g )0 (0 ≤ j ≤ n − are pairwise distinct factors of the above direct product M ∗ . n upper bound for the nonsolvable length Then, for every non-trivial z ∈ W , there exist u ∈ h zH i and b ∈ ( A ∗ ) k such thatthe n + 1 elements ( u ⊗ w j ( bg ) (0 ≤ j ≤ n ) are pairwise distinct. Here, the number of choices for b is at least | A ∗ | k / | A | .Proof. For convenince of notation we let T = { t , . . . , t m } with t = 1 and M i = M t i and A i = A t i for all i . Without loss we may assume that A w j ( g )0 = A j for0 ≤ j ≤ n −
1. Clearly, A j acts non-trivially on the block W i = W ⊗ t i if and onlyif i = j .Suppose firstly, that A w n ( g )0 = A j for 0 ≤ j ≤ n −
1. Then, for every u ∈ W ,the vectors ( u ⊗ w j ( g ) (0 ≤ j ≤ n ) belong to pairwise different blocks and are,therefore, all distinct. In this situation any k -tuple b ∈ ( A ∗ ) k gives rise to a sequence (cid:8) ( u ⊗ t i ) w j ( bg ) (cid:9) nj =0 with n + 1 distinct elements. Since the number of such tuples b is | A ∗ | k , the claim holds.Suppose next, that there exists some ℓ ∈ { , . . . n − } such that A w n ( g )0 = A w ℓ ( g )0 = A ℓ . Modulo application of a suitable Nielsen transformation to the variables x , . . . , x k we may assume without loss, that y n = x k is the last letter occurring in the word w . We focus our attention on the word µ = w − ℓ w = y ℓ +1 · · · y n and on its partialsubwords µ j = w − ℓ w ℓ + j = y ℓ +1 · · · y ℓ + j for 0 ≤ j ≤ n − ℓ .For every j ∈ { , . . . , n − ℓ } and every b ∈ ( A ∗ ) k , we define elements f j ( b ) ∈ M ∗ by the equation f j ( b ) µ j ( g ) = µ j ( bg ) . Each µ j ( g ) is in N G ( A ∗ ) and, using Lemma 1, is readily seen that each f j ( b ) belongsto A ∗ . Fix any non-trivial vector z ∈ W and consider z ⊗ t ℓ ∈ W ℓ = W w ℓ ( g ).Choose now b , . . . , b k − freely in A ∗ . This choice can be done in | A ∗ | k − differentways. The choice of b k will be described just below.In order to express some relations in a clearer fashion, we will interpret elements d ∈ M ∗ as functions d : { , . . . , m } −→ M with values d [ i ] (0 ≤ i ≤ m ). Notice that W ℓ µ ( g ) = W ℓ . We need to choose b k ∈ A ∗ in such a way that ( z ⊗ t ℓ ) µ ( bg ) = z ⊗ t ℓ .To this end, we choose b k under the only constrain that(1) ( z ⊗ t ℓ ) µ ( bg ) = ( z ⊗ t ℓ ) f n − ℓ − ( b ) µ n − ℓ − ( g )( b k g k ) = z ⊗ t ℓ . Note that( z ⊗ t ℓ ) f n − ℓ − ( b ) µ n − ℓ − ( g )( b k g k ) = ( z ⊗ t ℓ ) f n − ℓ − ( b ) b µ n − ℓ − ( g ) − k µ ( g )= z (cid:16) f n − ℓ − ( b ) b µ n − ℓ − ( g ) − k (cid:17) [ ℓ ] ⊗ t ℓ µ ( g )= v ⊗ t ℓ where v = zf n − ℓ − ( b )[ ℓ ]( b k [ ℓ.µ n − ℓ − ( g )]) τ h , for some automorphism τ of A andfor some h ∈ H defined by t l µ ( g ) = ht l . Inequality (1) holds if and only if(2) f n − ℓ − ( b )[ ℓ ]( b k [ ℓ.µ n − ℓ − ( g )]) τ h Stab H ( z ) . The calculation of f n − ℓ − ( b )[ ℓ ] does not need the knowledge of the element b k [ ℓ.µ n − ℓ − ( g )], as can be checked easily from the identities of Lemma 1 and the fumagalli | leinen | puglisi fact that, for all 0 ≤ i < j < n − l the subgroups A lµ i ( g ) and A lµ j ( g ) are distinct.We let s = f n − ℓ − ( b )[ ℓ ] and show that there exists some element c ∈ A such that(3) sc τ h Stab H ( z ) . If c has been found, then the element b k can be defined by setting b k [ ℓ.µ n − ℓ − ( g )] = c , while b k [ i ] can be chosen freely for all i = ℓ.µ n − ℓ − ( g ).If z = zs = zh − = z , then the coset { y ∈ H | z y = z } of Stab H ( z ) isdistinct from Stab H ( z ). Therefore, this coset cannot contain any subgroup. Inparticular, it cannot contain A τ . In this situation, we can select c ∈ A such, thatthe inequality (3) holds, and the choice of b k is completed.We are left with the situation when zs = zh − . Note that z was an arbitrarynon-trivial element from W up to now. Pick r ∈ H and consider u = zr . If us = uh − we replace z by u and apply the above argument with u instead of z .Thus, it remains to study the case when ( zr ) s = ( zr ) h − for all r ∈ H . Since A acts non-trivially on W = h zH i , there exist certain r ∈ H and c ∈ A suchthat ( zr ) sc τ = ( zr ) s = ( zr ) h − and the choice of b k can again be completed using u = zr in place of z .Using the chosen element b k , we now have( zr ⊗ t ℓ ) µ ( bg ) = zrsc τ h ⊗ t ℓ = zr ⊗ t ℓ for suitable r ∈ H , and the elements ( u ⊗ w ( bg ) , . . . , ( u ⊗ w n ( bg ) are all distinctfor u = zr . Because the element b k could be chosen freely in all but one of itscomponents, the possible choices for b k are at least | A ∗ | / | A | , hence the possiblechoices for the k -tuples b satisfying the requirements of the Lemma are at least | A ∗ | k / | A | . (cid:3) We will also need the following fact.
Lemma 4.
Let G ≤ Sym(Ω) be a finite transitive group with a non-trivial normalsubgroup M ∗ = Q mi =0 M i , where the subgroups M i are permuted transitively by G under the action by conjugation. Suppose that ω ∈ Ω , and let R be a subgroup of H = N G ( M ) containing ( G ω ∩ H ) c M , where c M = Q i =0 M i .If W is an irreducible constituent of Ind HR ( C ) , on which M acts non-trivially,then C Ω has a submodule isomorphic to V = Ind GH ( W ) .Proof. By Lemma 2 the module V = Ind GH ( W ) is irreducible. To prove the claim itis therefore enough to show that Hom G ( C Ω , V ) = 0. By Frobenius reciprocity([4,Theorem 10.8]), we haveHom G ( C Ω , V ) ≃ Hom G ω ( C , V | G ω ) . The module V is itself an induced module so, by an application of Mackey’s theorem([4, Theorem 10.13]), we have that V | G ω contains, as a direct summand, the G ω -module Ind G ω H ∩ G ω ( W ). A further application of Frobenius reciprocity entailsHom G ω (Ind G ω H ω ( W ) , C ) ≃ Hom H ω ( W, C ) , and Hom H ω ( W, C ) = 0 by our choice of W . (cid:3) n upper bound for the nonsolvable length Lemma 5.
Let G ≤ Sym(Ω) be a finite transitive group and ω any point of Ω . If V is a non-trivial G -submodule of C Ω and π is the projection from C Ω onto V ,then ( ω ) π = 0 .Proof. Write C Ω = V ⊕ U with U G -invariant. If ( ω ) π = 0 then ω ∈ U andtherefore ωg ∈ U for every g ∈ G . Hence Ω ⊆ U because G is transitive on Ω. Itfollows that U = C Ω and V = 0, a contradiction. (cid:3) m -rarefied groups and strategy of the proof For every finite group G we define the following characteristic subgroups: R ( G ) = h A | A is a normal solvable subgroup of G i ,S ( G ) = h B | B is a minimal normal non-abelian subgroup of G i . The RS -series of G is then defined recursively by R ( G ) = R ( G ) S i ( G ) /R i ( G ) = S ( G/R i ( G )) , for i ≥ R i +1 ( G ) /S i ( G ) = R ( G/S i ( G )) , for i ≥ . The series thus defined always reaches G . If G = R m +1 ( G ) > R m ( G ), then m issaid to be the nonsolvable length of G and we write λ ( G ) = m .By [6, Theorem 1.1], every group G of nonsolvable length m has an m - rarefied subgroup, that is, a subgroup H of nonsolvable length m satisfying the followingconditions:(i) R ( H ) = Φ( H ) and R i +1 ( H ) /S i ( H ) = Φ( H/S i ( H )) for all i = 1 , . . . , m − S i ( H ) /R i ( H ) is the unique minimal normal subgroup of H/R i ( H ) for1 ≤ i ≤ m ;(iii) the simple components of S i ( H ) /R i ( H ) are isomorphic to groups in the set L = (cid:8) L (2 r ) , L (3 r ) , L ( p a ) , L (3) , B (2 r ) (cid:12)(cid:12) p, r odd primes , a ∈ N (cid:9) for 1 ≤ i ≤ m . Lemma 6.
Let S be a simple group in L . Then S contains an Aut( S ) -invariant S -conjugacy class of subgroups, which are dihedral of order p for some odd prime p .Proof. We first claim that S contains a conjugacy class of (maximal) subgroupsthat are dihedral of order 2 m , for some odd integer m .When the characteristic of S is 2, choose the conjugacy class of normalizers ofsplit-torii subgroups (diagonal subgroups) of order q −
1. Each of these subgroupsis a dihedral group of order 2( q −
1) (see [7, Theorems 6.5.1 and 6.5.4]).When S ≃ L ( q ), with q = 3 r or q = p a ( p, r odd primes and a ≥ S contains two conjugacy classes of maximal subgroups, which are dihedral subgroupsof orders respectively q − q + 1. Each of these subgroups is the normalizer fumagalli | leinen | puglisi respectively of a split torus (diagonal subgroup of S ) or of a non-split torus. Ac-cording to the congruence class of q modulo 4, take the class of subgroups havingorder 2 m , with m odd (see [7, Theorem 6.5.1]).When S = L (3) the normalizers of the Sylow 13-subgroups form a conjugacyclass of dihedral groups of order 26 (see [3]).Finally, consider one of the dihedral groups E in the conjugacy class just estab-lished. If | E | = 2 m , then choose a prime p dividing m . The group E has a uniquesubgroup C of order p . And if a is any involution in E , then D = C h a i is a dihedralgroup of order 2 p . Since all the involutions of E are conjugate in E , any othersubgroup of E of order 2 p is conjugate to D . The set { D g | g ∈ S } is then a familyof subgroups with the required properties. (cid:3) Theorem C will be proved by contradiction. So assume, that there exists a group G with nonsolvable length n , such that the conclusion of Theorem C is not validwith respect to the action of G on a certain faithful transitive G -set Ω. Among allpossible such counterexamples, we select a pair ( G, Ω) such that n + | G | + | Ω | isminimal.Consider an n -rarefied subgroup H of G . From [6, Proposition 4.7], we obtain λ ( H/C H (Γ)) = n for at least one H -orbit Γ in Ω. Hence, by minimal choice of( G, Ω), we have G = H , that is, G is an n -rarefied group. By [6, Proposition4.2], every proper subgroup U of G satisfies λ ( U ) ≤ n . Hence minimality of ( G, Ω)yields, that every proper subgroup of G has nonsolvable length strictly smaller than n . 4. Proof of Theorem C: reduction to the case Φ( G ) = 1 . In this section, ( G , Ω ) is always a minimal counterexample to TheoremC, as above. F = Φ( G ) denotes the Frattini subgroup of G , and L denotes the lastterm of the derived series of S ( G ). Since G is n -rarefied, S ( G ) /F is the uniqueminimal normal subgroup in G/F . In particular, S ( G ) = F L . Lemma 7.
Assume that K is a normal subgroup of G not containing L . Then K ≤ F and G/K is n -rarefied.Proof. Suppose that K ∩ L is not contained in F . Then ( K ∩ L ) F/F is a non-trivialnormal subgroup of
G/F and therefore it contains S ( G ) /F , the unique minimalnormal subgroup of G/F . Thus S ( G ) = ( K ∩ L ) F and L = L ∩ S ( G ) = L ∩ ( K ∩ L ) F = ( L ∩ F )( K ∩ L ) . Since L is perfect, we get L = L ′ ≤ ( L ∩ F ) ′ ( K ∩ L ). Because L ∩ F is nilpotent,iteration of this argument finally yields L = K ∩ L , that is, L ≤ K . But thiscontradicts the hypothesis of the Lemma. Hence K ∩ L ≤ F , and this impliesthat KF/F centralizes
LF/F = S ( G ) /F . On the other hand S ( G ) /F has trivialcentralizer in G/F (by [6, Lemma 2.4]). It follows that K ≤ F . Hence G/K is n -rarefied by [6, Proposition 4.2 and Lemma 4.3]. (cid:3) n upper bound for the nonsolvable length Lemma 8. If G ω < H ≤ G , for some ω ∈ Ω , then L ≤ H G . Moreover L lies inthe normalizer of any block system (with respect to the action of G on Ω ).Proof. Let H = { Hg | g ∈ G } . The translation action of G on H has kernel H G ,and |H| < | Ω | . Assume, that H G does not contain L . Then, by Lemma 7, G/H G is n -rarefied. And since | G/H G | + |H| < | G | + | Ω | , the group G/H G satisfies TheoremC. For each k -variable word w of length n , we can thus find a Sylow 2-subgroup Q/H G and y ∈ ( Q/H G ) k and α ∈ H , such that (cid:12)(cid:12) { αw i ( y ) | ≤ i ≤ n } (cid:12)(cid:12) = n + 1.Because G/H G acts transitively on H , there is no loss in assuming that α = H .Note that Q/H G = P H G /H G for some Sylow 2-subgroup P in G . Let x ∈ P k bea preimage to y . Since the points αw i ( y ) (0 ≤ i ≤ n ) are pairwise distinct, theproducts w i ( x ) w j ( x ) − (0 ≤ i < j ≤ n ) do not lie in H and not in G ω . But thenthe points ωw i ( x ) (0 ≤ i ≤ n ) are pairwise distinct too, showing that G satisfiesTheorem C, a contradiction.Let K be the normalizer in G of the block system B . Note that KG ω is asubgroup. Assume, that K = 1. So G acts faithfully on B and, since |B| < | Ω | , wehave that G satisfies the conclusion of Theorem C in its action on B . It follows,that G satisfies the conclusion of Theorem C in its action on Ω, a contradiction.Now K = 1, whence KG ω > G ω . We conclude, that L is contained in the core of KG ω , which equals K . (cid:3) Proposition 1.
The Frattini subgroup F of G is a p -group for some prime p , andthe subgroup A = F ∩ L is abelian.Proof. We only need to consider the case when F = 1. Let us show firstly, that thecentralizer C F ( L ) is trivial.To this end, assume that D = C F ( L ) = 1. Let K denote the normalizer of theblock system B consisting of the D -orbits in Ω. By Lemma 8, L ≤ K . Consider a D -orbit ∆ ∈ B . By [5, Theorem 4.2A], the centralizer C of D/C D (∆) in Sym(∆)is isomorphic to a section of D/C D (∆), hence nilpotent. Because the actions of D and L on ∆ commute, the group L/C L (∆) is isomorphic to a subgroup of C ,hence nilpotent. On the other hand, L is perfect. It follows that L = C L (∆).Since this happens for every orbit ∆, we obtain L = 1. But this is impossible,because λ ( G ) = n ≥
1. Hence, D = 1. It follows immediately, that the intersection A = L ∩ F is non-trivial.We will show now, that A is a p -group. To this end, assume that the order of A is divisible by two different primes p and q . Recall, that A is a normal subgroupin G , because F and L are normal subgroups in G . Moreover, A is nilotent. Let P resp. Q be the Sylow p - resp. the Sylow q -subgroup of A . Then P and Q arenormal subgroups of G . Since the stabilizer G ω of a point ω ∈ Ω has trivial core, P and Q cannot be contained in G ω . Application of Lemma 8 to QG ω in the role of H yields L ≤ QG ω , and Dedekind’s modular law gives A = A ∩ QG ω = Q ( G ω ∩ A ).It follows that P ≤ G ω ∩ A , a contradiction to P G ω .Now A is a p -group for some prime p . Assume next, that F is not a p -group.Then F has non-trivial q -subgroup Q for some prime q = p . Again, Q is normal in fumagalli | leinen | puglisi G and QG ω > G ω . Therefore, L ≤ QG ω and A = F ∩ L ≤ F ∩ QG ω = Q ( F ∩ G ω ).It follows that A ≤ F ∩ G ω , a contradiction to A G ω .It remains to show, that A is abelian. Consider a group B , which is normalizedby G ω and satisfies F ω < B ≤ F . Lemma 8 shows, that L ≤ BG ω , whence A = L ∩ BG ω ∩ F = L ∩ B ( G ω ∩ F ) = L ∩ BF ω = L ∩ B ≤ B. Assume, that there exist two distinct minimal such groups B and B . This wouldimply A ≤ B ∩ B = F ω ≤ G ω , in contradiction to G ω having trivial core in G .We conclude, that for each ω ∈ Ω, there is a unique minimal subgroup B ( ω ), whichis normalized by G ω and satisfies F ω < B ( ω ) ≤ F .Because F is nilpotent, the group K ( ω ) = N F ( F ω ) is strictly larger than F ω .Moreover, K ( ω ) is normalized by G ω . Thus, B ( ω ) is contained in K ( ω ). It follows,that the subgroup A of B ( ω ) normalizes F ω too. Now AF ω is a subgroup of B ( ω ),which is normalized by G ω . Since A is not contained in F ω , we obtain AF ω = B ( ω ).The normal subgroup V = T ω ∈ Ω B ( ω ) of G contains A . And so it suffices toshow, that V is abelian. Consider a maximal subgroup M of B ( ω ) containing F ω .Because F ω is normalized by G ω , the G ω -core M = T g ∈ G ω M g of M contains F ω .By minimality of B ( ω ), we have M = F ω . Since F is nilpotent, the commutatorsubgroup (cid:0) B ( ω ) (cid:1) ′ of B ( ω ) is contained in the G ω -conjugates of M and hence in M = F ω . It follows that V ′ ≤ T ω ∈ Ω (cid:0) B ( ω ) (cid:1) ′ ≤ T ω ∈ Ω F ω = 1 . (cid:3) We shall show now that the Frattini subgroup F is even trivial. Proposition 2.
The group G has trivial Frattini subgroup.Proof. Assume that F = 1. In the sequel, p will denote the prime dividing theorder of F (Proposition 1). We prove firstly, that L splits over A = L ∩ F and that S ( G ) splits over F .Let B = Φ( L ), assume B = 1, and consider any B -orbit Γ in Ω. If Γ = Ω wouldhold, we could apply the Frattini argument to obtain G = BG ω for every ω ∈ Ω.But this is clearly impossible, because B ≤ F and F consists of non-generators for G ([14, 5.2.12]). It follows, that the B -orbits form a proper block system in the G -set Ω. By Lemma 8, L normalizes every B -orbit Γ. In particular, the Frattiniargument yields L = BL γ for any point γ ∈ Γ, and this is again impossible. Thiscontradiction shows, that B = 1.Applying Proposition 1 and [14, Theorem 5.2.13], we obtain that L splits overthe abelian normal subgroup A . A complement K to A in L is a complement to F in S ( G ), so that S ( G ) splits over F too. Because S ( G ) /F is a minimal normalsubgroup in G/F , there exists a simple group S in L such that K = Q ℓi =1 S i whereeach S i is a copy of S . In the sequel, we will distinguish the two cases when p isodd and when p = 2, but try to treat them simultaneously.When p > , let R be any Sylow 2-subgroup of K . Clearly R = Q ℓi =1 R i , whereeach R i is a Sylow 2-subgroup of S i . When p = , Lemma 6 gives us an Aut( S )-invariant S -conjugacy class of dihedral subgroups of order 2 r in S , for some oddprime r . Let D i denote the copy of this conjugacy class in S i . We can now choose a n upper bound for the nonsolvable length subgroup U i ∈ D i for each i ∈ { , . . . , ℓ } and define U = Q ℓi =1 U i and R = Q ℓi =1 R i where R i is the Sylow r -subgroup in U i . Next step.
Our next aim is it to show, that if R ≤ G ω , then K ≤ G ω .Assume by way of contradiction, that R ≤ G ω and S i G ω for some i . Withoutloss, i = 1. Because the simple group S is generated by the conjugates of R in S , there exists s ∈ S such that R s G ω . Let Q = R s and Q i = R s i for all i . Clearly Q i = R i when i = 1. Therefore, Q i ≤ G ω if and only if i = 1. When p = , we also let D = U s and D i = U is for all i .In the case when p > , an application of Frattini argument within the group G/F yields
G/F = ( S ( G ) /F )( Y /F ), where
Y /F denotes the normalizer of theSylow 2-subgroup
QF/F in G/F . For every y ∈ Y , the Sylow 2-subgroup Q y of QF is conjugate to Q by a suitable element f ∈ F , and yf − ∈ N G ( Q ). This showsthat G = S ( G ) X for X = N G ( Q ).In the case when p = , the G -conjugates of DF/F are conjugate in S ( G ) /F .Therefore, the Frattini argument yields G = S ( G ) N G ( DF ). Because Q is a Sylow r -subgroup of DF , it is possible to apply the Frattini argument again. It follows,that N G ( DF ) = F DX , where X = N N G ( F D ) ( Q ) ≥ D . Altogether, G = S ( G ) X .Conclusion: G = S ( G ) X where (cid:26) X = N G ( Q ) for p > X = N G ( Q ) ∩ N G ( F D ) for p = 2Since S ( G ) /F is minimal normal in G/F , the factorization G = S ( G ) X entails,that X acts transitively on the set { S i F | i = 1 , . . . , ℓ } by conjugation. Consider x ∈ X and indices i, j such that ( S i F ) x = S j F . Then Q xi ≤ Q ∩ S j F ≤ K ∩ S j F = S j ( K ∩ F ) = S j , whence Q xi ≤ Q ∩ S j = Q j . This shows, that X also acts tran-sitively on the set { Q i | i = 1 , . . . , ℓ } by conjugation. The group X/ ( X ∩ S ( G )) is( n − G/S ( G ). Moreover, X < G , because Q is not normal in G . Altogether, λ ( X ) = n − X = N X ( Q ) as the stabilizer of Q under the conjugation action of X on { Q i | i = 1 , . . . , ℓ } . In the case when p > , we let Z = X ∩ X ω . Clearly, Q Z because Q X ω . In the case when p = , we let Z = ( X ∩ X ω )( F ∩ X ). In orderto see, that Z does not contain Q also in this case, we argue by contradiction:Assume that Q ≤ Z . Then ωQ ⊆ ωZ = ω ( X ∩ X ω )( F ∩ X ) = ω ( F ∩ X ) . The group X ∩ F centralizes Q , because [ Q, X ∩ F ] ≤ Q ∩ F = 1. Therefore, Q is normal in Z and ωQ is a block under the action of Z on ωZ . It follows, that r = | ωQ | divides | ω ( F ∩ X ) | = | ( F ∩ X ) : ( F ∩ X ω ) | . But the later is a power of2, a contradiction.We have now shown that Q Z in all cases.However, Z contains the normal subgroup c Q = Q ℓi =2 Q i of X . Hence we canchoose an irreducible constituent W of the C X -module Ind X Z ( C ), on which Q acts non-trivially. Note that c Q acts trivially on W , because it acts trivially on the fumagalli | leinen | puglisi whole Ind X Z ( C ). Since the normal subgroup Q of X acts non-trivially on W , itsfixed point space C W ( Q ) is a proper X -submodule of W , whence C W ( Q ) = 0.We choose a right transversal T = { t , t , . . . , t ℓ } of X in X such that t = 1and Q t i = Q i for all i . Then V = Ind XX ( W ) = L ℓi =1 ( W ⊗ X t i ) . We applyLemma 2 with
X, X , Q, Q in the roles of G, H, h M G i , M (resp.) and obtain,that V is an irreducible C X -module. The subspaces W i = W ⊗ X t i are blocksunder the action of X , and each factor Q i acts non-trivially only on W i . Note that C W ( Q ) = 0 entails C W i ( Q i ) = 0 for all i .Let Γ = ωX and consider the permutation module C Γ. We apply Lemma 4with
X, X , Z, Q, Q in the roles of G, H, R, h M G i , M (resp.). Note that thehypotheses of the lemma are satisfied, because ( G ω ∩ X ) c Q ≤ X ω ∩ X ≤ Z . Itfollows, that C Γ contains V as a direct summand.Consider the image v = P ℓi =1 ( v i ⊗ t i ) of ω under the projection of C Γ onto V . Then v = 0 by Lemma 5. If b ∈ Q j for some j ≥
2, then vb = v because Q j ≤ G ω . On the other hand, every W i ( i = j ) is fixed pointwise by Q j , whence( v i ⊗ t i ) b = v i ⊗ t i . Thus, v = vb = (cid:16) ℓ X i =1 v i ⊗ t i (cid:17) b = (cid:16) X i = j ( v i ⊗ t i ) (cid:17) + ( v j ⊗ t j ) b . We conclude, that v j ⊗ t j = ( v j ⊗ t j ) b for all b ∈ Q j and that v j ⊗ t j ∈ C W j ( Q j ) = 0.Hence v ∈ W .Consider the block system B = { W i | ≤ i ≤ ℓ } in the C X -module W . For every x ∈ N X ( B ) and every j ∈ { , . . . , ℓ } , the group Q xj fixes L i = j W i pointwise. It fol-lows, that Q xj = Q j for every j . In particular, N X ( B ) normalizes every componentof S ( G ) /F , whence N X ( B ) S ( G ) /S ( G ) is solvable. But then, N X ( B ) ≤ R ( G )and λ ( X/N X ( B )) = λ ( X ) = n − w ∈ F ∞ be any reduced k -variable word of length n . Again, we distinguishour two cases. Case 1 . p > . With respect to its action on B , the group X/N X ( B ) satisfies Theorem C. Hencethere exist a Sylow 2-subgroup P in X and x ∈ P k such that W w i ( x ) = W w j ( x ),for all 0 ≤ i < j ≤ n −
1. An application of Lemma 3 with
X, X , Q, Q in theroles of G, H, M ∗ , A ∗ (resp.) yields an element u ∈ vX and a k -tuple b ∈ Q k such that uw i ( bx ) = uw j ( bx ) for all 0 ≤ i < j ≤ n . Because the 2-group Q isnormal in X , we have Q ≤ P and bg ∈ P k . If u = vg , with g ∈ X , then u is theprojection of δ = ωg on V . Hence δw i ( bx ) = δw j ( bx ), for all 0 ≤ i < j ≤ n . Butthen, G would not be a counterexample to Theorem C. This contradition shows,that R ≤ G ω implies K ≤ G ω in the case when p is odd. Case 2 . p = . Since Q E D and X = N G ( Q ) ∩ N G ( DF ), we have [ D, X ] ≤ DF ∩ N G ( Q ) = DF ∩ X = D ( F ∩ X ). In particular, D ( F ∩ X ) E X . Pick a Sylow 2-subgroup J in D . Then J = J ( X ∩ F ) is a Sylow 2-subgroup in D ( X ∩ F ), and the Frattini n upper bound for the nonsolvable length argument gives X = D ( X ∩ F ) Y , where Y = N X ( J ). But D ( X ∩ F ) ≤ S ( G ), andso G = S ( G ) X = S ( G ) Y .Clearly, J = Q ℓi =1 J i for certain Sylow 2-subgroups J i of D i (1 ≤ i ≤ ℓ ).Moreover, Y must permute the subgroups J i ( X ∩ F ) (1 ≤ i ≤ ℓ ) transitively viaconjugation, as well as the subgroups S i F (1 ≤ i ≤ ℓ ). Therefore, it follows as abovewith J i ( X ∩ F ) in place of Q i , that Y acts transitively on B and that Y /N Y ( B ) hasnon-solvable length n − Y /N Y ( B ) satisfies Theorem C, and there exist a Sylow 2-subgroup P of Y and x ∈ P k such that W w i ( x ) = W w j ( x ), for all 0 ≤ i < j ≤ n −
1. Let Y = N Y ( J ) = N Y (cid:0) J ( X ∩ F ) (cid:1) . Note, that the intersection X ∩ F acts triviallyon V , because it is a normal subgroup of X with trivial action on W . Therefore wecan apply Lemma 3 with the quotients of the groups Y, Y , J, J modulo ( X ∩ F )in the roles of G, H, M ∗ , A ∗ (resp.). This yields an element u ∈ vY and a k -tuple b ∈ J k such that uw i ( bx ) = uw j ( bx ) for all 0 ≤ i < j ≤ n .Because the 2-group J is normal in Y , we have J ≤ P and bg ∈ P k . If u = vg ,with g ∈ Y , then u is the projection of δ = ωg on V . Hence δw i ( bx ) = δw j ( bx ),for all 0 ≤ i < j ≤ n . But then, G would not be a counterexample to Theorem C.This contradition shows, that R ≤ G ω implies K ≤ G ω in the case when p = 2 too.Conclusion: R ≤ G ω implies K ≤ G ω for all p .Since the normal subgroup F of G is not contained in G ω , Lemma 8 gives S ( G ) = F L ≤ F G ω and S ( G ) = F ( S ( G ) ∩ G ω ) for every ω ∈ Ω. Since R is a 2-groupsresp. an r -group, a conjugate of R is contained in a Sylow subgroup of S ( G ) ∩ G ω .Therefore R ≤ G α for some α ∈ Ω, and hence K ≤ G α as well.Clearly R is normal in N = N G ( R ), and so R ≤ G β and K ≤ G β for every β ∈ αN . In particular, K is contained in the intersection H = \ β ∈ αN ( S ( G ) ∩ G β ) . In particular, 1 = K ≤ H < S ( G ) and H is normalized by N . Consider M = HN .An application of the Frattini argument to the subgroup R of S ( G ) shows, that G = S ( G ) N . Since S ( G ) = F K and K ≤ M , we have that G = F M . However, F consists of non-generators for the group G . Therefore G = M . But this impliesΩ = αHN = αN and 1 = H ≤ T β ∈ αN G β = T ω ∈ Ω G ω = 1, a contradiction. (cid:3) Proof of the main theorems
We are now well-prepared to take the final steps in the proof of Theorem C.
Theorem C.
Let G be a finite group with λ ( G ) ≥ n , and let Ω be a faithfultransitive G -set. Then, for every ω ∈ Ω and for every non-trivial reduced word w = w ( x , . . . , x k ) ∈ F ∞ of length n , there exist a Sylow -subgroup P of G anda tuple g ∈ P k such that the points ωw ( g ) , ωw ( g ) , . . . , ωw n ( g ) are pairwisedistinct. In particular, G ∈ P n . fumagalli | leinen | puglisi Proof.
Assume, that Theorem C is wrong, and consider a minimal counterexample( G, Ω) with λ ( G ) = n . By Proposition 2, the unique minimal normal subgroup S ( G ) in G is the product Q j ∈ ∆ S j of copies S j of a simple group S ∈ L . Let∆ = { , . . . , ℓ } . We also write S ∗ instead of S ( G ) and let π j : S ∗ −→ S j denotethe canonical projection. Lemma 6 gives us an Aut( S )-invariant S -conjugacy classof dihedral subgroups of order 2 p in S , for some odd prime p . Let D j denote thecopy of this conjugacy class in S j .We aim to show firstly, that G = S ∗ N G ( D ∗ ) for a certain subgroup D ∗ of S ∗ ,which is the direct product of dihedral groups D j ∈ D j (1 ≤ j ≤ ℓ ). Case 1.
For any ω ∈ Ω, the subgroup G ω ∩ S ∗ is not a subdirect product of thecomponents S j .Without loss we assume that ( G ω ∩ S ∗ ) π < S . Then it is possible to find D ∈ D such that the Sylow p -subgroup C of D is not contained in ( G ω ∩ S ∗ ) π . Because D is generated by its involutions, there is also a Sylow 2-subgroup Q ≤ D , whichis not contained in ( G ω ∩ S ∗ ) π . For every j = 1 we choose D j ∈ D j arbitrarilyand form D ∗ = Q j ∈ ∆ D j . The G -conjugates of D ∗ are precisely the S ∗ -conjugatesof D ∗ . Therefore the Frattini argument yields G = S ∗ N G ( D ∗ ). Case 2.
For any ω ∈ Ω, the subgroup G ω ∩ S ∗ is a subdirect product of thecomponents S j .In this case, there exists a partition U = { ∆ i | i = 1 , . . . , m } of ∆ such that G ω ∩ S ∗ = Q mi =1 U i where each U i is a diagonal subgroup of S (∆ i ) = Q j ∈ ∆ i S j .Here, some of the ∆ i must contain at least two points. Without loss we mayassume, that | ∆ | ≥ containing 1. The conjugation action of G ω on G ω ∩ S ∗ gives rise to an action of G ω on the set U , and G ω normalizes O = { ∆ i ∈ U | | ∆ i | ≥ } .Assume, that the action on O is not transitive, and consider a G ω -orbit I ⊆ O .Because we are in case 2, it is possible to choose I such that the G ω -invariantsubgroup S ( I ) = Q ∆ i ∈I Q j ∈ ∆ i S j is not contained in G ω . From Lemma 8, themonolith S ∗ is contained in S ( I ) G ω , and Dedekind’s identity yields S ∗ = S ∗ ∩ S ( I ) G ω = ( S ∗ ∩ G ω ) S ( I ). Since I 6 = O , there exists ∆ ∈ O \ I . But now, theprojection of S ∗ = ( S ∗ ∩ G ω ) S ( I ) on S (∆) is isomorphic to S , a contradiction to | ∆ | ≥
2. We conclude, that all the ∆ i of size ≥ G ω .In particular, they all have the same size d .There are automorphisms σ , . . . , σ d ∈ Aut( S ) such that U = { ( s, s σ , . . . , s σ d ) | s ∈ S } . For each j ∈ ∆ , we can choose a subgroup D j ∈ D j , such that U ∩ (cid:0) Q j ∈ ∆ D j (cid:1) =1 . We can proceed in the same fashion with every ∆ i ∈ O . When S i ≤ G ω , wechoose D i ∈ D i freely. Let D ∗ = Q i ∈ ∆ D i . Note, that D ∗ ∩ G ω is the product ofprecisely those factors D j , for which S j ≤ G ω . In particular, ( D ∗ ∩ G ω ) π = 1.And also in this case, the Frattini argument gives G = S ∗ N G ( D ∗ ).We proceed to work with the group D ∗ constructed in the two cases. For each j ∈ ∆ choose a Sylow 2-subgroup Q j of D j (in case 1, the group Q has already n upper bound for the nonsolvable length been choosen), and let C j be the Sylow p -subgroup of D j . As usual, we let C ∗ = Q j ∈ ∆ C j and Q ∗ = Q j ∈ ∆ Q j . Further applications of the Frattini argument yield G = S ∗ N G ( D ∗ ) and N G ( D ∗ ) = D ∗ N N G ( D ∗ ) ( Q ∗ ), whence G = S ∗ X for X = N G ( D ∗ ) ∩ N G ( Q ∗ ).It follows, that X must act transitively via conjugation on the set { D j | j ∈ ∆ } .Moreover, X < G and
G/S ∗ ≃ X/ ( X ∩ S ∗ ) imply that λ ( X ) = n −
1. Considerthe subgroup A = C ∗ X and note, that A contains D ∗ = C ∗ Q ∗ . We observe, thatany non-trivial element c from C ∗ with non-trivial constituent in some C j can beconjugated by an involution in D j , whence C j is contained in the normal closureof c in A . Together with the transitivity of the action of X on the set { D j | j ∈ ∆ } this yields, that C ∗ is a minimal normal subgroup in A . Now C is not containedin G ω . It follows, that C ∗ acts faithfully on the set Γ = ωA . Consider the kernel K of the action of A on Γ. Since K ∩ C ∗ = 1, the subgroup K commutes with C ∗ .But then K normalizes every C j and hence every S j . It follows that K ≤ R ( G )and λ ( A/K ) = λ ( A ) = n − K would contain an involution a from D ∗ , then the subgroup [ a, D ∗ ] of K would contain one of the C j . This argument shows, that D ∗ ∩ K = 1. Consider H = N A ( D ) and R = ( H ∩ A ω ) c D , where c D = Q i =1 D i . Assume, that C ≤ R .Then C ≤ D ∗ ∩ R ≤ ( D ∗ ∩ G ω ) b D . But this implies 1 = C ≤ ( D ∗ ∩ G ω ) π , acontradiction. It follows, that C R . In particular, there exists then an irreduciblecomponent W in the C H -module Ind HR ( C ) on which C acts non-trivially.From Lemmata 2 and 4, the C A -module V = W ⊗ H C A is (isomorphic to) asubmodule of C Γ. Clearly, V = ⊕ t ∈ T W t with W t = W ⊗ t , where T denotes atransversal of H in A . Since C ∗ ≤ H , the transversal can be chosen such that1 ∈ T ⊂ X . Note that X normalizes Q ∗ . Therefore, each index j ∈ ∆ determinesa unique t ∈ T such that D j = D t , and even C i = C t and Q i = Q t . Moreover, D j acts non-trivially on W t if and only if D j = D t , because b D acts trivially on W .Let v = P t ∈ T v t ⊗ t denote the projection of a point τ ∈ Γ on V . Then v = 0by Lemma 5, and we may choose τ in such a way, that v = 0. Consider anyreduced k -variable word w ∈ F ∞ of length n . The group G permutes the set { S j | j ∈ ∆ } of components of S ∗ . Because G is a minimal counterexample, thereexist a Sylow 2-subgroup P of X and a k -tuple g ∈ P k such that the components S w ( g ) , S w ( g ) , . . . , S w n − ( g ) are pairwise distinct. In particular, the subgroups D w ( g ) , D w ( g ) , . . . , D w n − ( g ) are pairwise distinct as well.Clearly, W = h v H i . And since D = h Q C i acts non-trivially on W , the group Q acts non-trivially on W too. We can therefore apply lemma 3 with A, H, D , Q in place of G, H, M , A (resp.). It follows, that there are a k -tuple b ∈ Q ∗ k and avector u = v h ∈ v H , such that the elements ( u ⊗ w j ( bg ) are pairwise distinctfor 0 ≤ j ≤ n . Here, bg ∈ ( Q ∗ P ) k and Q ∗ P is contained in a Sylow 2-subgroup P of G .The projection u = vh = P t ∈ T u t ⊗ t of τ h on V has non-trivial component u ⊗ W . We now refine our choice of T by asking that T contains a transversalof Z = H ∩ P in P . Consider the elements x ij = w i ( bg ) w j ( bg ) − for 0 ≤ i < j ≤ n . fumagalli | leinen | puglisi Let x be one of the elements x ij ∈ P . Then u ⊗ = ( u ⊗ x = u z ⊗ t , where x = zt for unique z ∈ Z and t ∈ T ∩ P . Since Z is a 2-group, | u Z | is a power of 2.Assume that u t C ⊆ u Z . The set u t C is a block for the action of H on u t H and therefore it is also a block for the action of Z on u t Z = u Z . In particular,also | C : Stab C ( u t ) | = | u t C | is a power of 2. On the other hand, the p -group C is a normal subgroup of H , which acts non-trivially on W , so that it must in factact fixed-point-freely on W , with orbits of size p . This contradiction shows, that u t C u Z .For each t ∈ T \{ } , we choose an element s t ∈ C such that u t s t / ∈ u Z . And welet s = 1. Consider s ∗ = Q t ∈ T s tt ∈ Q t ∈ T C t = C ∗ and the point α = τ hs ∗ ∈ Γ.The projection of α on V is˜ v = us ∗ = u ⊗ X = t ∈ T u t s t ⊗ t. We claim that ˜ vx ij = ˜ v for 1 ≤ i < j ≤ n .To this end we only need to show, that for every choice of i < j there exists some t ∈ T such that the W t -components of ˜ vx ij and ˜ v are different. In the case when x ij normalizes W , the W -component of ˜ vx ij is ( u ⊗ x ij = u ⊗
1. In the casewhen W x ij = W t = W , the W t -component of ˜ vx ij has the form y ⊗ t for some t ∈ u Z , while the W t -component of ˜ v is ( u t s t ) ⊗ t = y ⊗ t .We have shown now, that αx ij = α resp. αw i ( bg ) = αw j ( bg ) for all 0 ≤ i Consider any faithful transitive G -set Ω. By Theorem C,the group G satisfies P n on Ω. Thus we can find ω ∈ Ω and g ∈ G k such that ωw ( g ) = ωw n ( g ) = ω . This implies immediately, that w ( g ) = 1. ✷ Proof of Theorem A. Consider a non-trivial k -variable reduced word w ∈ F ∞ of length n = λ ( G ). Theorem B provides a k -tuple g ∈ H k such that w ( g ) = 1.Therefore, w is not a law in G . This shows, that λ ( G ) < ν ( G ). ✷ References [1] Meenaxi Bhattacharjee. The ubiquity of free subgroups in certain inverse limits of groups. J.Algebra , 172(1):134–146, 1995.[2] A. Bors and A. Shalev. Words, permutations, and the nonsolvable length of a finite group. arXiv:1904.02370v1 , 2019.[3] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson. Atlas of finitegroups . Oxford University Press, Eynsham, 1985. Maximal subgroups and ordinary charactersfor simple groups, With computational assistance from J. G. Thackray.[4] Charles W. Curtis and Irving Reiner. Methods of representation theory. Vol. I . Wiley ClassicsLibrary. John Wiley & Sons, Inc., New York, 1990. With applications to finite groups andorders, Reprint of the 1981 original, A Wiley-Interscience Publication.[5] John D. Dixon and Brian Mortimer. Permutation groups , volume 163 of Graduate Texts inMathematics . Springer-Verlag, New York, 1996. n upper bound for the nonsolvable length [6] Francesco Fumagalli, Felix Leinen, and Orazio Puglisi. A reduction theorem for nonsolvablefinite groups. Israel J. Math. , 232(1):231–260, 2019.[7] Daniel Gorenstein, Richard Lyons, and Ronald Solomon. The classification of the finite simplegroups. Number 3. Part I. Chapter A , volume 40 of Mathematical Surveys and Monographs .American Mathematical Society, Providence, RI, 1998. Almost simple K -groups.[8] Gareth A. Jones. Varieties and simple groups. J. Austral. Math. Soc. , 17:163–173, 1974.Collection of articles dedicated to the memory of Hanna Neumann, VI.[9] Evgenii I. Khukhro and Pavel Shumyatsky. Nonsoluble and non-p-soluble length of finitegroups. Israel J. Math. , 207(2):507–525, 2015.[10] Michael Larsen and Aner Shalev. Word maps and Waring type problems. J. Amer. Math.Soc. , 22(2):437–466, 2009.[11] Michael Larsen, Aner Shalev, and Pham Huu Tiep. The Waring problem for finite simplegroups. Ann. of Math. (2) , 174(3):1885–1950, 2011.[12] Felix Leinen and Orazio Puglisi. Free subgroups of inverse limits of iterated wreath productsof non-abelian finite simple groups in primitive actions. J. Group Theory , 20(4):749–761,2017.[13] Martin W. Liebeck, E. A. O’Brien, Aner Shalev, and Pham Huu Tiep. The Ore conjecture. J. Eur. Math. Soc. (JEMS) , 12(4):939–1008, 2010.[14] Derek J. S. Robinson. A course in the theory of groups , volume 80 of Graduate Texts inMathematics . Springer-Verlag, New York, second edition, 1996.[15] Aner Shalev. Some results and problems in the theory of word maps. In Erd¨os centennial ,volume 25 of Bolyai Soc. Math. Stud. , pages 611–649. J´anos Bolyai Math. Soc., Budapest,2013. Dipartimento di Matematica e Informatica “U. Dini”, Universit`a di Firenze, VialeMorgagni 67A, I-50134 Firenze, Italy Email address : francesco.fumagalli @ unifi.it Institute of Mathematics, Johannes Gutenberg-University, D − Email address : Leinen @ uni-mainz.de Dipartimento di Matematica e Informatica “U. Dini”, Universit`a di Firenze, VialeMorgagni 67A, I-50134 Firenze, Italy Email address ::