A generalization of Sims conjecture for finite primitive groups and two point stabilizers in primitive groups
aa r X i v : . [ m a t h . G R ] F e b A GENERALIZATION OF SIMS CONJECTURE FOR FINITE PRIMITIVE GROUPS AND TWOPOINT STABILIZERS IN PRIMITIVE GROUPS
PABLO SPIGA
Abstract.
In this paper we propose a refinement of Sims conjecture concerning the cardinality of the point stabilizers infinite primitive groups and we make some progress towards this refinement.In this process, when dealing with primitive groups of diagonal type, we construct a finite primitive group G on Ω andtwo distinct points α, β ∈ Ω with G αβ E G α and G αβ = 1, where G α is the stabilizer of α in G and G αβ is the stabilizer of α and β in G . In particular, this example gives an answer to a question raised independently by Peter Cameron in [3] andby Alexander Fomin in the Kourovka Notebook [14, Question 9.69]. Introduction
Let G be a finite primitive group acting on a set Ω and let α ∈ Ω. The subdegrees of G are the lengths of the orbitsof the point stabilizer G α := { g ∈ G | α g = α } on Ω.Given a subdegree d of G , there is no bound on the degree | Ω | of G , as a function of d only . For example, for anyprime p , the dihedral group of order 2 p has a faithful primitive permutation representation of degree p with subdegree d = 2. Despite this, if a finite primitive group G has a small subdegree, then the structure of G α is rather restricted, seefor instance [15, 20, 23] and the much more recent results in [7, 12]. Following the investigations on the cases d = 3 and d = 4, Charles Sims [20] was lead to conjecture the following. Theorem 1.1 (Sims conjecture) . There is a function f : N → N such that, if G is a finite primitive group with a suborbitof length d > , then the stabilizers have order at most f ( d ) . This theorem was proved by Cameron, Praeger, Saxl and Seitz [5] using the O’Nan-Scott Theorem and the recently (atthe time) announced classification of finite simple groups. The CFSGs spurred a new vitality in the old subject of finitepermutation groups and this result was one of the first major applications of the CFSGs, see also [4].In this paper we propose a strengthening of Sims conjecture that in our opinion captures the structure of point stabilizersof primitive groups to a finer degree. As well as Sims conjecture, our conjecture can be phrased in purely group-theoreticterminology, but it is better understood borrowing some terminology from graph theory.Let G be a finite primitive group acting on a set Ω and let α and β be two elements of Ω. The orbital graph Γdetermined by the ordered pair ( α, β ), is the directed graph with vertex set Ω and with arc set ( α, β ) G := { ( α g , β g ) | g ∈ G } .Clearly, G is a group of automorphisms of Γ acting primitively on its vertex set and Γ is undirected if and only if the orbital ( α, β ) G is self-paired , that is, ( α, β ) G = ( β, α ) G . We denote by Γ + ( γ ) := { δ | ( γ, δ ) ∈ ( α, β ) G } and byΓ − ( γ ) := { δ ∈ Ω | ( δ, γ ) ∈ ( α, β ) G } the out-neighborhood and the in-neighborhood , respectively, of the vertex γ of Γ.As Ω is finite, it follows that the out-valency and the in-valency of Γ are equal, that is, | Γ + ( γ ) | = | Γ − ( γ ) | for every γ ∈ Ω. Moreover, if we denote by d this valency, then d equals the cardinality of the suborbit β G α = Γ + ( α ). In particular, d is a subdegree of G .Given two vertices α and β as above, we write G αβ := G α ∩ G β . Moreover, we denote with G +[1] α := \ δ ∈ Γ + ( α ) G αδ and G − [1] β := \ δ ∈ Γ − ( β ) G δβ , the kernel of the action of G α on Γ + ( α ) and of G β on Γ − ( β ), respectively. Observe that the notation G +[1] α and G − [1] β is slightly misleading because it does not show the dependency of this subgroup of G from the graph Γ, however for notmaking the notation too cumbersome to use we prefer not to attach the label “Γ” in the notation for G +[1] α and G − [1] β .Furthermore, we denote by G Γ + ( α ) α ∼ = G α /G +[1] α and by G Γ − ( β ) β ∼ = G β /G − [1] β , the permutation group induced by G β onΓ + ( α ) and on Γ − ( β ), respectively. The groups G Γ + ( α ) α and G Γ − ( β ) β are (not necessarily isomorphic) permutation groupsof degree d and they are sometimes refereed to as the local groups , see [16, 17] where this terminology is particularlysuited.Using the notation that we have established above, Sims conjecture claims that, when d >
1, the cardinality of G α isbounded above by a function of the valency d of the orbital graph Γ. In other words, the order of the vertex stabilizer G α Mathematics Subject Classification.
Primary 20B25; Secondary 05E18.
Key words and phrases. arc-transitive graph, primitive group, Sims Conjecture, subdegree. is bounded above by a function of the local group G Γ + ( α ) α . (Actually, the order of the vertex stabilizer is bounded abovesimply by a function of the degree of the local group.) Broadly speaking, we wish to make a step further and we wonderwhether, besides some families that can be explicitly classified, the order of the arc stabilizer G αβ is bounded above bya function of the order of the point stabilizer G Γ + ( α ) αβ of the local group G Γ + ( α ) α . More precisely, we propose the followingconjecture: Conjecture 1.2.
There exists a function g : N → N such that, if G is a finite primitive group with a suborbit β G α oflength d > , then either (i): G αβ has order at most g ( | G αβ : G +[1] α | ) , or (ii): G is in a well-described and well-determined list of exceptions. It is quite unfortunate, that we cannot omit alternative ( ii ) in Conjecture 1.2. Examples in this direction are intricateto construct and we give an example later in this paper, see Example 4.2. A positive solution to Conjecture 1.2 can be seenas a strengthening of Sims conjecture, this is easy to see but we postpone the proof to Section 2.1. However, a positiveanswer to Conjecture 1.2 is a stronger result than Sims conjecture: we refer the reader to Example 2.2 to see this and tocapture the idea behind our question. In this paper, we give some evidence towards the veracity of Conjecture 1.2.A particularly interesting case for our conjecture is when G αβ E G α . In this case, G +[1] α = G αβ , that is, the local group G Γ + ( α ) α is regular. In this case, it was asked by Peter Cameron [3] whether G αβ = 1, that is, whether the whole vertexstabilizer G α acts regularly on the out-neighborhood Γ + ( α ). This question was also proposed by Fomin in the KourovkaNotebook [14, Question 9.69]. Remarkable evidence towards the veracity of the question of Cameron and Fomin is givenby Konygin [8, 9, 10, 11], however to the best of our knowledge this question is still open. To some extent, Conjecture 1.2can be seen as a generalization of the question of Cameron and Fomin, allowing G αβ E G α , but relaxing the conclusionwhen G αβ E G α . By investigating Conjecture 1.2 for primitive groups of diagonal type, we construct a finite primitivegroup with G αβ E G α and G αβ = 1, thus giving an answer in the negative to the question of Cameron and Fomin. Wepresent this example in Section 5. 2. Basic results
Relations between Sims conjecture and Conjecture 1.2.Lemma 2.1.
Suppose that Conjecture . holds true. Then, any group satisfying part ( i ) of Conjecture . , satisfies Simsconjecture.Proof. Let g be the function arising from a positive solution of Conjecture 1.2. We define a function g ′ : N → N by setting g ′ ( d ) := max { g ( | H δ | ) | H transitive on { , . . . , d } , δ ∈ { , . . . , d }} . Now, let G be a finite primitive group satisfying part (i) of Conjecture 1.2, let β G α be a suborbit of G of cardinality d > H := G +Γ( α ) α and observe that H is a transitive permutationgroup on Γ + ( α ) of degree d . The stabiliser of a point in H is G Γ + ( α ) αβ . Since G satisfies Conjecture 1.2 (i) , we have | G αβ | ≤ g ( | G Γ + ( α ) αβ | ) ≤ g ′ ( d ) . Now, | G α | = | G α : G αβ || G αβ | = d | G αβ | ≤ dg ′ ( d ). Therefore, Sims conjecture holds for G by taking f ( d ) := dg ′ ( d ). (cid:3) Example 2.2.
It is interesting to consider finite primitive groups having a suborbit β G α of odd cardinality d with G α acting as a dihedral group on β G α . In this case, | G α : G αβ | = d and | G αβ : G +[1] α | = 2. It follows from the main resultin [19] (and also from the work of Verret on p -subregular actions [21, 22]), that | G αβ | divides 16. In particular, | G αβ | ≤ | G αβ | does not depend on d .In particular, in this example, Sims conjecture requires bounding | G α | as a function of d . However, Conjecture 1.2 ismore demanding: since | G αβ : G +[1] α | = 2 is a constant, Conjecture 1.2 demands either regarding G is an exception orbounding | G αβ | from above with an absolute constant. Luckily, in this case, G is not an exception, because from [21, 22], | G αβ | ≤ (ii) .2.2. The O’Nan-Scott theorem and our investigation.
The modern key for analyzing a finite primitive permutationgroup G is to study the socle N of G , that is, the subgroup generated by the minimal normal subgroups of G . The socleof an arbitrary finite group is isomorphic to the non-trivial direct product of simple groups; moreover, for finite primitivegroups these simple groups are pairwise isomorphic. The O’Nan-Scott theorem describes in details the embedding of N in G and collects some useful information about the action of N . In [13, Theorem] five types of primitive groups aredefined (depending on the group- and action-structure of the socle), namely HA ( Affine ), AS (
Almost Simple ), SD (
SimpleDiagonal ), PA (
Product Action ) and TW (
Twisted Wreath ), and it is shown that every primitive group belongs to exactly
WO POINT STABILIZERS 3 one of these types. We remark that in [18] this subdivision into types is refined, namely the PA type in [13] is partitionedin four parts, which are called HS (
Holomorphic simple ), HC (
Holomorphic compound ), CD (
Compound Diagonal ) andPA. For what follows, we find it convenient to use this subdivision into eight types of the finite primitive primitive groups.In this paper we investigate Conjecture 1.2 using the O’Nan-Scott theorem.3.
Primitive groups of HA, TW, HS and HC type
Proposition 3.1.
Let G be a transitive group on Ω containing a normal regular subgroup, let Γ be a connected digraphwith vertex set Ω and left invariant by the action of G . Then G +[1] α = 1 , for every α ∈ Ω . In particular, Conjecture . i ) holds true when G has O’Nan-Scott type HA , TW , HS and HC and the function g can be taken so that g ( n ) := 1 , forevery n ∈ N .Proof. Let α ∈ Ω. Let N be a regular normal subgroup of G and let H := G α . Then, G is the semidirect product of N by H (that is, G = N G α , N ∩ G α = 1 and G = N ⋊ H ) and the action of G on Ω is permutation equivalent to the “affine”action of G on N , where N acts on N by right multiplication and where H acts on N by conjugation. In the rest of theproof, we use this identification. In particular, under this equivalence, α ∈ Ω corresponds to 1 ∈ N . For each β ∈ Γ + ( α ),we let n β be the element of N corresponding to β , that is, α n β = β .Then G αβ = C H ( n β ) and G +[1] α = \ β ∈ Γ + ( α ) G αβ = \ β ∈ Γ + ( α ) C H ( n β ) = C H ( h n β | β ∈ Γ + ( α ) i ) . Since Γ is connected, we deduce N = h n β | β ∈ Γ + ( α ) i . Hence G +[1] α = C H ( N ) = 1. In other words, G α acts faithfully onΓ + ( α ).When G has O’Nan-Scott type HA, TW, HS and HC, the proof follows by the fact that G contains a normal regularsubgroup and by the fact that the non-trivial orbital graphs of G are connected. (cid:3) Primitive groups of SD type
We start by recalling the structure of the finite primitive groups of SD type. This will also allow us to set up thenotation for this section.Let ℓ ≥ T be a non-abelian simple group. Consider the group N = T ℓ +1 and D = { ( t, . . . , t ) ∈ N | t ∈ T } , adiagonal subgroup of N . Set Ω := N/D , the set of right cosets of D in N . Then | Ω | = | T | ℓ . Moreover we may identifyeach element ω ∈ Ω with an element of T ℓ as follows: the right coset ω = D ( α , α , . . . , α ℓ ) contains a unique elementwhose first coordinate is 1, namely, the element (1 , α − α , . . . , α − α ℓ ). We choose this distinguished coset representative.Now the element ϕ of Aut( T ) acts on Ω by D (1 , α , . . . , α ℓ ) ϕ = D (1 , α ϕ , . . . , α ϕℓ ) . Note that this action is well-defined because D is Aut( T )-invariant. Next, the element ( t , . . . , t ℓ ) of N acts on Ω by D (1 , α , . . . , α ℓ ) ( t ,...,t ℓ ) = D ( t , α t , . . . , α ℓ t ℓ ) = D (1 , t − α t , . . . , t − α ℓ t ℓ ) . Observe that the action induced by ( t, . . . , t ) ∈ N on Ω is the same as the action induced by the inner automorphismcorresponding to the conjugation by t . Finally, the element σ in Sym( { , . . . , ℓ } ) acts on Ω simply by permuting thecoordinates. Note that this action is well-defined because D is Sym( ℓ + 1)-invariant.The set of all permutations we described generates a group W isomorphic to T ℓ +1 · (Out( T ) × Sym( ℓ + 1)) . A subgroup G of W containing the socle N of W is primitive if either ℓ = 2 or G acts primitively by conjugation on the ℓ + 1 simple direct factors of N , see [6, Theorem 4.5A]. The group G is said to be primitive of SD type, when the secondcase occurs, that is, N E G ≤ W and G acts primitively by conjugation on the ℓ + 1 simple direct factors of N .Write M = { ( t , t , . . . , t ℓ ) ∈ N | t = 1 } . Clearly, M is a normal subgroup of N acting regularly on Ω. Since the stabilizer in W of the point D (1 , . . . ,
1) isSym( ℓ + 1) × Aut( T ), we obtain W = (Sym( ℓ + 1) × Aut( T )) M. Moreover, every element x ∈ W can be written uniquely as x = σϕm , with σ ∈ Sym( ℓ + 1), ϕ ∈ Aut( T ) and m ∈ M . Theorem 4.1.
Let G be a primitive group on Ω of SD type, let α and β be two distinct elements from Ω and considerthe action of G on the orbital graph determined by ( α, β ) . Then one of the following holds • G +[1] α = 1 , • | G +[1] α | = ℓ + 1 , G +[1] α ≤ Sym( ℓ + 1) and G +[1] α acts regularly on { , . . . , ℓ + 1 } . Moreover, let ( t , t , . . . , t ℓ ) ∈ N with t = 1 and β = D ( t , t , . . . , t ℓ ) . The mapping G +[1] α → T defined by σ → t σ − is a group homomorphism. PABLO SPIGA
Proof.
We use the notation that we have established above. Without loss of generality we may assume that α = D (1 , , . . . , . Write β := D (1 , t , . . . , t ℓ ) , for some t , . . . , t ℓ ∈ T . We set t := 1, in particular, β = D ( t , t , . . . , t ℓ ). This notation will make the last part of ourproof easier to follow.Let ϕ ∈ G +[1] α ∩ Aut( T ). For each t ∈ T , we let ι t ∈ Aut( T ) ≤ W α denote the permutation on Ω induced by theconjugation via t . Observe that ι t ∈ G ∩ W α = G α , for every t ∈ T , because T ℓ +1 = N ≤ G . As ϕ ∈ G +[1] α and ι t ∈ G α ,we deduce that ϕ fixes β ι t , for every t ∈ T . This means that(1) D (1 , t t , . . . , t tℓ ) = β ι t = β ι t ) ϕ = D (1 , t t , . . . , t tℓ ) ϕ = D (1 , t tϕ , . . . , t tϕℓ ) , for every t ∈ T . Since β = α , there exists i ∈ { , . . . , ℓ } with t i = 1. Now, (1) gives t tϕi = t ti , for every t ∈ T . Therefore ϕ ∈ \ t ∈ T C Aut( T ) ( t ti ) = C Aut( T ) ( h t ti | t ∈ T i ) = C Aut( T ) ( T ) = 1 . This shows that(2) G +[1] α ∩ Aut( T ) = 1 . Now, G +[1] α is a normal subgroup of G α . Since G α acts primitively as a group of permutations on the ℓ + 1 simple directfactors of T ℓ +1 , we obtain that either G +[1] α projects trivially on Sym( ℓ + 1) or G +[1] α projects to a transitive subgroup ofSym( ℓ + 1). If G +[1] α projects trivially on Sym( ℓ + 1), then G +[1] α ≤ Aut( T ) and hence G +[1] α = 1 by (2). (In particular, inthis case Conjecture 1.2 part (i) holds true). Therefore, for the rest of this proof we assume that G +[1] α projects to a transitive subgroup of Sym( ℓ + 1).Observe now that ι ( T ) = { ι t | t ∈ T } ≤ G α (because T ℓ +1 = N ≤ G ) and that ι ( T ) ⊳ G α (because W α = Aut( T ) × Sym( ℓ + 1) and G α ≤ W α ). As G +[1] α ∩ Aut( T ) = 1, we deduce G +[1] α ∩ ι ( T ) = 1. Since G +[1] α and ι ( T ) are both normal in G α , we deduce that G +[1] α centralizes ι ( T ). Since C W α ( ι ( T )) = Sym( ℓ + 1), we get G +[1] α ≤ Sym( ℓ + 1). Therefore G +[1] α is a transitive subgroup of Sym( ℓ + 1).Next we show that G +[1] α is a regular subgroup of Sym( ℓ + 1). Let H := N G ( T ), where T is the first simple directfactor of the socle N . Now, H acts transitively on Ω, because N ≤ H , and H contains the normal regular subgroup M = T × · · · × T ℓ . Therefore, by Proposition 3.1 applied to H , we deduce H +[1] α = 1. As | G : H | = | G : N G ( T ) | = ℓ + 1and H +[1] α = H ∩ G +[1] α , we get | G +[1] α | ≤ ℓ + 1. Since G +[1] α is a transitive subgroup of Sym( ℓ + 1), G +[1] α is a regularsubgroup of Sym( ℓ + 1) and ℓ + 1 = | G +[1] α | . We need to recall in detail the action of Sym( ℓ + 1) on Ω. Given σ ∈ Sym( ℓ + 1) and ω = D ( x , x , . . . , x ℓ ) ∈ Ω, wehave(3) ω σ = D ( x σ − , x σ − , . . . , x ℓ σ − ) . The element σ in the right hand side of (3) appears as σ − to guarantee that this is a right action.Recall β = D (1 , t , . . . , t ℓ ) = D ( t , t , . . . , t ℓ ). We now define a mapping w : G +[1] α T,σ t σ − . In other words, in the light of (3), w ( σ ) is the first coordinate of ( t , t , . . . , t ℓ ) σ = ( t σ − , t σ − , . . . , t ℓ σ − ).Let σ, τ ∈ G +[1] α . Since τ fixes β , we have β = β τ = D ( t τ − , t τ − , . . . , t ℓ τ − )and since στ fixes β , we have also β = β στ = D ( t ( στ ) − , t ( στ ) − , . . . , t ℓ ( στ ) − ) . In other words, the two ( ℓ + 1)-tuples ( t τ − , t τ − , . . . , t ℓ τ − ) and ( t ( στ ) − , t ( στ ) − , . . . , t ℓ ( στ ) − ) differ only by the leftmultiplication by an element of D . Therefore, there exists t ∈ T such that(4) ( tt τ − , tt τ − , . . . , tt ℓ τ − ) = ( t ( στ ) − , t ( στ ) − , . . . , t ℓ ( στ ) − ) . By checking the first coordinates in (4), we obtain tt τ − = t ( στ ) − . WO POINT STABILIZERS 5
Moreover, by comparing the coordinate appearing in position 0 τ on the left-hand side and on the right-hand side of (4),we deduce tt (0 τ ) τ − = t (0 τ ) ( στ ) − , that is, t = tt = t σ − . Putting these two equations together, we obtain w ( σ ) w ( τ ) = t σ − t τ − = tt τ − = t ( στ ) − = w ( στ ) . This proves that our mapping w : G +[1] α → T is a group homomorphism. (cid:3) The proof of Proposition 4.1 hints to the fact that in Conjecture 1.2 we do need the alternative (ii) . We show that thisis indeed the case in the next example.
Example 4.2.
Let p be a prime number, let k ≥ r be a primitive prime divisor of p k − r is relatively prime to p i − i ∈ { , . . . , k − } . As k ≥
7, the existence of r is guaranteed by Zsigmondy’stheorem.Let H := V ⋊ C be the affine primitive group of degree p k , where V is an elementary abelian p -group of order p k andwhere R is a cyclic group of order r . (We use an additive notation for V .) Let T be a non-abelian simple group containinga cyclic subgroup P of order p and with C T ( P ) = P and let w : V → P be an arbitrary surjective homomorphism.We denote by T V the set of all functions from V to T . Observe that T V is a group isomorphic to the Cartesian productof | V | = p k copies of T . We denote the elements of T V as functions f : V → T .We let G be primitive group of diagonal type T V ⋊ H. Recall that the elements of Ω are right cosets of D in T V , where D is the diagonal subgroup of T V , that is, D = { f ∈ T V | f is constant } . Let b = ( t v ) v ∈ V ∈ T V where t v = ω ( − v ), for every v ∈ V .Let α := D , let β := Db and consider the orbital graph determined by ( α, β ). Let v ∈ V . Then b v ( x ) = b ( x − v ) = w ( − x + v ) = w ( − x ) w ( v ) = w ( v ) w ( − x ) = w ( v ) b ( x ) , ∀ x ∈ V. Thus b v = w ( v ) b and hence β v = Db v = Db = β . This shows V ≤ G αβ . Since V E G α , we deduce G +[1] α ≤ V . Now, fromProposition 4.1, we have G +[1] α = V .Thus G α = ( T × R ) G +[1] α and G αβ = ( G αβ ∩ ( T × R )) G +[1] α . Let ϕ := th ∈ G αβ ∩ ( T × R ), with t ∈ T and h ∈ R . Observe that(5) b th (0) = b (0 h − ) t = b (0) t = w (0) t = 1 t = 1 = w (0) = b (0) . Since Db = β = β th = Db th = Db , from (5) we get b th = b .For every v ∈ Ker( w ) ≤ V , we have w ( − v h − ) t = ( b ( v h − )) t = b th ( v ) = b ( v ) = w ( − v ) = 1 . Therefore, w ( v h − ) = 1. This gives Ker( w ) h − = Ker( w ). As dim Ker( w ) = k − = 0 and as R is a cyclic group of primeorder acting irreducibly on the vector space V , we deduce h = 1. This shows that ϕ = t ∈ T .Now, b t = b if and only if t centralizes all the coordinates of b . In other words, t ∈ C T ( P ) = P . Summing up, G αβ = P × V and G +[1] α = V. Thus | G αβ : G +[1] α | = | P | = p and | G +[1] α | = p k . However, we cannot bound the cardinality of G +[1] α with p only.5. The example for the Cameron and Fomin question
Our construction is quite elaborate and requires a number of ingredients: • let A be a non-abelian simple group, • let T be a non-abelian simple group containing a subgroup Q with Q = A × A and with C T ( Q ) = 1, • let H be a group containing A with A maximal in H , A core-free in H and, in the faithful permutation action of H on the right cosets of A , there exist two points whose setwise stabilizer is the identity.We observe that there are groups A, T, Q and H satisfying the hypothesis above. For instance, we may take A := Alt(5), T := Alt(10), Q = Alt(5) × Alt(5) ≤ T and H := PSL ( p ) where p is a prime number with p ≥
61 and p ≡ ± A and T are non-abelian simple and C T ( Q ) = 1. Moreover, using the hypothesis on p , A is a maximal subgroupin the Aschbacher class S of T , see for instance [1, Table 8 . p >
19, we see from [2, Table 1] thatthe base size of T in the action on the right cosets of A is 2. Actually, from the arguments in [2], it follows that whenever p ≥
61, there exist two points whose setwise stabilizer is the identity.
PABLO SPIGA
Let V := A | H : A | and let L := V ⋊ H be a primitive group of TW type with regular socle V and with point stabilizer H . The fact that L is primitive in its action on V follows from [6, Lemma 4 . T V the set of all functions from V to T . We let G be primitive group of diagonal type T V ⋊ L. The elements of Ω are right cosets of D in T V , where D is the diagonal subgroup of T V , that is, D = { f ∈ T V | f is constant } .Relabelling the elements in the domain { , . . . , | H : A |} , we may suppose that 1 , H and thesetwise stabilizer of { , } in H is the identity. We define the group homomorphism w : V = A | H : A | Q ≤ T, ( a , a , . . . , a | H : A | ) ( a , a ) . Let b ∈ T V with b ( v ) = ω ( v − ), for every v ∈ V . Let α := D , let β := Db and consider the orbital graph determinedby ( α, β ). Let v ∈ V . Then b v ( x ) = b ( xv − ) = w (( xv − ) − ) = w ( vx − ) = w ( v ) w ( x − ) = w ( v ) b ( x ) , ∀ x ∈ V. Thus b v = w ( v ) b and hence β v = Db v = Db = β . This shows V ≤ G αβ . Since V E G α , we deduce G +[1] α ≤ V . Now, fromProposition 4.1, we have G +[1] α = V .Thus G α = T × L = T × HV = T × HG +[1] α = ( T × H ) G +[1] α and(6) G αβ = ( G αβ ∩ ( T × H )) G +[1] α . Let ϕ := th ∈ G αβ ∩ ( T × H ), with t ∈ T and h ∈ H . Observe that(7) b th (1) = b (1 h − ) t = b (1) t = w (1) t = (1 , t = (1 ,
1) = w (1) = b (1) . Since Db = β = β th = Db th = Db , from (7) we get b th = b .For every v = (1 , , a , . . . , a | H : A | ) ∈ Ker( w ) ≤ A | H : A | = V , we have w (( v − ) h − ) t = ( b ( v h − )) t = b th ( v ) = b ( v ) = w ( v − ) = (1 , . Therefore, w (( v − ) h − ) = (1 , v = (1 , , a , . . . , a | H : A | ) ∈ Ker( w ) ≤ A | H : A | = V . This gives that h fixes setwise { , } and hence h = 1, by our assumption on the permutation action of H on the right cosets of A . This shows that ϕ = t ∈ T .Now, b t = b if and only if, for every v = ( a , a , . . . , a | H : A | ) ∈ V , we have b t ( v ) = b ( v ), that is, ( b ( v )) t = b ( v ). Thisyields ( a − , a − ) t = ( w ( v − )) t = ( b ( v )) t = b ( v ) = w ( v − ) = ( a − , a − ) . Since this holds for each ( a , a ) ∈ A × A = Q ≤ T , we deduce t ∈ C T ( Q ) = 1. This shows that ϕ = 1.As ϕ was an arbitrary element in G αβ ∩ ( T × H ), we get G αβ ∩ ( T × H ) = 1 and hence G αβ = G +[1] α from (6). Therefore G αβ E G α . References [1] J. N. Bray, D. F. Holt, C. M. Roney-Dougal,
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