aa r X i v : . [ m a t h . G R ] F e b BASES OF TWISTED WREATH PRODUCTS
JOANNA B. FAWCETT
Dedicated to the memory of my doctoral supervisor Jan Saxl
Abstract.
We study the base sizes of finite quasiprimitive permutation groups of twisted wreathtype , which are precisely the finite permutation groups with a unique minimal normal subgroupthat is also non-abelian, non-simple and regular. Every permutation group of twisted wreathtype is permutation isomorphic to a twisted wreath product G = T k : P acting on its base groupΩ = T k , where T is some non-abelian simple group and P is some group acting transitively on k = { , . . . , k } with k >
2. We prove that if G is primitive on Ω and P is quasiprimitive on k ,then G has base size 2. We also prove that the proportion of pairs of points that are basesfor G tends to 1 as | G | → ∞ when G is primitive on Ω and P is primitive on k . Lastly, wedetermine the base size of any quasiprimitive group of twisted wreath type up to four possiblevalues (and three in the primitive case). In particular, we demonstrate that there are manyfamilies of primitive groups of twisted wreath type with arbitrarily large base sizes. Introduction
Bases are a fundamental tool in permutation group theory and are used extensively in com-putational group theory (see [54]). For a permutation group G on a set Ω, a base is a subsetof Ω whose pointwise stabiliser in G is trivial. The base size of G , denoted by b Ω ( G ) or b ( G ),is the minimal cardinality of a base for G . It is immediate from the definition of a base that | G | | Ω | b ( G ) , so b ( G ) was first studied in order to bound the order of a finite primitive per-mutation group (e.g., [2, 7]). Increasingly, however, attention has focused on the parameter b ( G ) itself, and with the advent of the classification of the finite simple groups (CFSG), therehave been significant advances in our understanding of the base sizes of finite primitive groups,several of which are outlined below.This paper concerns bases of twisted wreath products. By the O’Nan-Scott theorem (see [41]or § § G is some semidirect product B : P with B ≃ T k such that T is a finite non-abelian simple group and P is a finite group acting transitivelyon k := { , . . . , k } where k >
2. The base group B is the unique minimal normal subgroupof G , so B = soc( G ), the socle of G . Further, G acts faithfully and transitively on B , with B acting regularly by right multiplication, and P acting by conjugation. In particular, G isquasiprimitive. (A permutation group is quasiprimitive if every non-trivial normal subgroup istransitive. For example, every primitive permutation group is quasiprimitive.) There are twomain issues when working with twisted wreath products. The first is that given some T and P as above, it can be difficult to determine whether a twisted wreath product G exists. Thesecond is that given some twisted wreath product G , the conditions on T and P for G to beprimitive are quite subtle; necessarily, the action of P on k is faithful, and T is a compositionfactor of P , the stabiliser in P of 1 ∈ k . Both issues are addressed in [4].One of the key advances in our understanding of base sizes concerns almost simple groups.Let C denote the class of finite almost simple primitive permutation groups, but exclude the standard actions, which, roughly speaking, are those arising from the alternating group A n on aset of subsets or partitions of n , or a classical group on a set of subspaces of the natural module Mathematics Subject Classification.
Key words and phrases. permutation group, base size, twisted wreath product, primitive, quasiprimitive. (see [9, Definition 2.1]). The base size of a standard permutation group can be arbitrarily large.However, Cameron conjectured [15] that there is an absolute constant c such that b ( G ) c for all G ∈ C . This conjecture was proved by Liebeck and Shalev [43], and, remarkably, it waslater established that c = 7 is best possible [9, 11, 13, 14]. Jan Saxl then initiated an ambitiousproject to classify the primitive permutation groups with base size 2. Much progress has beenmade on this problem for primitive groups of almost simple type [11, 12, 14, 35, 36, 47], diagonaltype [24] and affine type [26, 27, 31, 37–39]. Our first main result establishes that a large classof primitive twisted wreath products have base size 2.First we require some terminology. A finite quasiprimitive permutation group G is of twistedwreath type if G has a unique non-abelian non-simple regular minimal normal subgroup (inwhich case G is permutation isomorphic to a twisted wreath product by [48]). The top group of G is the permutation group induced by the conjugation action of G on the simple factors ofsoc( G ). If G is a twisted wreath product B : P as described above, then G has top group P k ,and if G is primitive on B , then P is faithful on k (see § Theorem 1.1.
Let G be a primitive permutation group of twisted wreath type whose top groupis quasiprimitive. Then b ( G ) = 2 . In fact, Theorem 1.1 is a consequence of a more technical result, Theorem 7.1, where we provethat many quasiprimitive permutation groups of twisted wreath type have base size 2.
Remark 1.2.
To provide some context for the assumption in Theorem 1.1 that the top group isquasiprimitive, we appeal to a result of Baddeley [4] (see Theorem 4.10). He proves that if G is aprimitive permutation group of twisted wreath type, then either G is of minimal-twisted type—inwhich case the top group is quasiprimitive—or G is a blow-up of a primitive permutation group H of minimal-twisted type, in which case soc( H ) r E G H ≀ S r where G acts on soc( H ) r viathe product action and r >
2. This gives us the following version of the O’Nan-Scott theorem:any primitive permutation group with a non-abelian socle is either of almost simple, diagonal orminimal-twisted type, or a blow-up of one of these types. While very little is known about theprimitive blow-ups with base size 2, much is known about the primitive groups of almost simpleor diagonal type with base size 2. Thus Theorem 1.1 contributes to this picture by showing thatevery primitive group of minimal-twisted type has base size 2. Moreover, we use Baddeley’sresult and Theorem 1.1 to prove the primitive case of Theorem 1.3 below.While we are unable to completely classify the primitive groups of twisted wreath type withbase size 2, we instead determine the base size of any quasiprimitive group of twisted wreathtype up to four possible values (and three in the primitive case). Our result is closely related toanother significant advance in the theory of base sizes. Recall that if G is a permutation group ofdegree n , then | G | n b ( G ) , so log n | G | b ( G ). Pyber conjectured [51] that there is an absoluteconstant c such that b ( G ) c log n | G | for every finite primitive group G of degree n . Building onthe work of [6, 8, 24, 28, 30, 43, 44, 46], this conjecture was recently established by Duyan, Halasiand Mar´oti [22]. Halasi, Liebeck and Mar´oti [32] then proved that b ( G ) n | G | )+ 24, whichis asymptotically best possible. One of the key ingredients in the proof of Pyber’s conjectureconcerns the distinguishing number of a permutation group G on Ω, which is the minimal numberof parts in a partition of Ω for which only the identity fixes every part; we denote it by d Ω ( G )or d ( G ). In [22], it is proved that if G is a transitive permutation group of degree n >
1, then n p | G | < d ( G ) n p | G | . We use this theorem to establish our next two results. Theorem 1.3.
Let G be a quasiprimitive permutation group on Ω of twisted wreath type withsocle T k and top group R , where T is a simple group. Then b Ω ( G ) = (cid:24) log | G | log | Ω | (cid:25) + ε = (cid:24) log d k ( R )log | T | (cid:25) + δ for some ε, δ ∈ { , , , } where ε δ , and if G is primitive on Ω , then ε = 3 and δ ε + 1 . ASES OF TWISTED WREATH PRODUCTS 3
For any non-abelian simple group T and transitive permutation group R of degree k >
2, thereis a quasiprimitive group of twisted wreath type with socle T k and top group R (see Remark 4.5).Since the symmetric group S r has distinguishing number r , Theorem 1.3 demonstrates that thebase size of a quasiprimitive group of twisted wreath type can be arbitrarily large. In fact, wewill see that this is also true for primitive groups of twisted wreath type.In our next result, we establish nearly exact base size formulae for a large class of primitivepermutation groups of twisted wreath type. Theorem 1.4.
Let H be a primitive permutation group on ∆ of twisted wreath type with socle T m , where T is a simple group. Let K be a transitive subgroup of S r where r > . Then G := H ≀ K is a primitive permutation group on Ω := ∆ r of twisted wreath type. If the top groupof H is quasiprimitive, then b Ω ( G ) = (cid:24) log | G | log | Ω | (cid:25) + ε = (cid:24) log d r ( K ) m log | T | (cid:25) + δ for some ε ∈ { , } and δ ∈ { , } where ε + 1 δ . Thus the base size of a primitive group of twisted wreath type can be arbitrarily large. Notethat for any non-abelian simple group T , there are primitive groups of twisted wreath type withquasiprimitive top groups and socle T m for infinitely many m (see Examples 9.1 and 9.3), so thebase size of a primitive group of twisted wreath type can grow in various ways (see Example 9.6).Our final result is related to a probabilistic version of the Cameron conjecture. Recall theclass C of almost simple groups that was defined above. Cameron and Kantor conjectured [16]that there is an absolute constant c ′ such that the proportion of c ′ -tuples of points that are(ordered) bases for G tends to 1 as | G | → ∞ for G ∈ C . This conjecture was proved by Liebeckand Shalev [43], and it was later shown that c ′ = 6 is best possible [10, 13, 45]. Analogousresults have been established for primitive groups of diagonal type [24] and coprime primitivelinear groups [23]. The following provides a partial result for the twisted wreath case. Theorem 1.5.
For those primitive permutation groups G of twisted wreath type whose top groupsare primitive, the proportion of pairs of points that are bases for G tends to 1 as | G | → ∞ . This paper is organised as follows. In §
2, we provide some basic notation and terminology,along with a brief description of the O’Nan-Scott theorem. In §
3, we describe some of the boundson distinguishing numbers from the literature. In §
4, we outline the theory of twisted wreathproducts and state a version of Baddeley’s classification of the primitive permutation groups oftwisted wreath type. Theorems 1.1, 1.4 and 1.5 are all consequences of more technical resultsthat concern much larger classes of groups. In §
5, we prove Theorem 1.5 and its more generalform. In §
6, we give some general bounds on the base size of a twisted wreath product. In § §
8, we prove Theorem 1.3 as well asTheorem 1.4 and its more general form. Lastly, we give some examples in § § § Preliminaries
All groups in this paper are finite, and all actions are written on the right. Basic terminologyfor permutation groups may be found in [20]. For a subgroup H of a group G , the core of H in G , denoted by core G ( H ), is defined to be ∩ g ∈ G H g ; it is precisely the kernel of the action of G on the (right) cosets of H . For a group G acting on a set Ω, we write G Ω for the permutationgroup induced by G on Ω. For g ∈ G , we write fix Ω ( g ) for the set of fixed points of g . We alsowrite ω Ω ( G ) for the number of orbits of G on Ω, and we write ω Ω ( g ) for ω Ω ( h g i ).Next we give some notation for wreath products and their actions. Let H be a permutationgroup on ∆, and let K S r . The wreath product H ≀ K is the semidirect product H r : K ,where π − ∈ K maps ( h , . . . , h r ) ∈ H r to ( h π , . . . , h r π ). In the product action of H ≀ K JOANNA B. FAWCETT on ∆ r , the elements ( h , . . . , h r ) ∈ H r and π − ∈ K map ( δ , . . . , δ r ) ∈ ∆ r to ( δ h , . . . , δ h r r )and ( δ π , . . . , δ r π ), respectively. In the imprimitive action of H ≀ K on ∆ × r , the element( h , . . . , h r ) π ∈ H ≀ K maps ( δ, i ) ∈ ∆ × r to ( δ h i , i π ).We now define the blow-up of a permutation group, following [4, § W := H ≀ S r , with H and r as above. Let ρ : W → S r be the projection map, and let W be the preimage of thestabiliser in S r of the point r ∈ r , so that W = ( H ≀ S r − ) × H . Let σ : W → H be theprojection map. A permutation group X is a blow-up of H on ∆ of index r if X is permutationisomorphic to a subgroup G of H ≀ S r containing soc( H ) r where G acts via the product action, ρ ( G ) is transitive on r and σ ( G ∩ W ) = H .To finish this section, we briefly consider the structure of a quasiprimitive permutation group.By the O’Nan-Scott theorem [41], the finite primitive permutation groups are of affine type,almost simple type, diagonal type, product type or twisted wreath type. A similar theorem isproved in [48] for the finite quasiprimitive permutation groups. In Table 1, we give a roughdescription of these types (using a similar format to the table in [8]). As in the primitive case,a quasiprimitive group G has at most two minimal normal subgroups, and when there are two,they are isomorphic and regular. The socle B of G is therefore isomorphic to T k for some simplegroup T and k >
1, and T is abelian if and only if G is of affine type. Note that in typeIII(b)(ii), the action of G is the usual product action, but this is not necessarily the case in typeIII(b)(i) when G is not primitive. Note also that in type II, the socle T may be regular when G is quasiprimitive, but T is not regular when G is primitive.Type DescriptionI Affine type: T abelianII Almost simple type: T G Aut( T )III(a)(i) Diagonal type: G T k . (Out( T ) × P ), P S k transitive, k > G T . Out( T ), k = 2III(b)(i) Product type: G H ≀ P , H quasiprimitive of type II, P S k transitive, k > G H ≀ P , H quasiprimitive of type III(a), P S ℓ transitive, ℓ | k III(c) Twisted wreath type: G = T k : P , P acts on k , k > Table 1.
The finite quasiprimitive groups G with socle T k Bounds on distinguishing numbers
In the introduction, we defined the distinguishing number of a permutation group. Since thisconcept is critical to our study of the base sizes of quasiprimitive groups of twisted wreath type,we give a brief survey of some of the known bounds on distinguishing numbers. For more detailsabout the origin of this parameter and its uses in graph theory, see the survey paper [5].First we need some terminology. Let G act on a set Ω. We say that a partition of Ω is distinguishing if the only elements of G that fix every part of the partition are those that fixevery element of Ω. The distinguishing number of G , denoted by d Ω ( G ) or d ( G ), is the minimalnumber of parts in a distinguishing partition for G . Similarly, we say that a subset of Ω is a base for G if the only elements of G that fix the subset pointwise are those that fix Ω pointwise, andwe denote the minimal size of a base for G by b Ω ( G ) or b ( G ). Observe that d ( G ) b ( G ) + 1.Observe also that d n ( S n ) = n and d n ( A n ) = n − G be a permutation group on Ω with degree n . Let P d (Ω) be the set of ordered partitionsof Ω with d (possibly empty) parts where d >
1. Since G has a regular orbit on P d ( G ) (Ω), itfollows that | G | < d ( G ) n when G = 1. The following remarkable result is [22, Theorem 1.2]. Theorem 3.1.
Let G be a transitive permutation group of degree n > . Then n p | G | < d ( G ) n p | G | . ASES OF TWISTED WREATH PRODUCTS 5
For certain classes of permutation groups, we can be much more precise. Let G be a permu-tation group on Ω with degree n , and suppose that G is not A n or S n . Cameron, Neumann andSaxl [17] proved that if G is primitive, then either G has a regular orbit on the power set of Ω(i.e., d ( G ) = 2), or G is one of finitely many exceptions. By work of Seress [53], the exceptionshave degree at most 32, and Dolfi later proved in [21, Lemma 1] that d ( G )
4. Moreover, theseresults were used by Duyan, Halasi and Mar´oti in their proof of Theorem 3.1 to show that if G is quasiprimitive, then d ( G )
4; see [22, Lemma 2.7]. In fact, Devillers, Morgan and Harperproved [19] that if G is quasiprimitive but not primitive, then d ( G ) = 2. They also provedthat if G is semiprimitive but not quasiprimitive, then d ( G )
3, with equality if and only if G = GL (3) in its action on non-zero vectors. (A permutation group G is semiprimitive if everynormal subgroup is transitive or semiregular.) Furthermore, Gluck proved in [29, Corollary 1]that if G has odd order, then d ( G )
2, and Seress proved in [52, Theorem 1.2] that if G issoluble, then d ( G ) A s is not a section of G for some s > d ( G ) s . We summarise these results in the following theorem for convenience. Theorem 3.2.
Let G be a non-trivial permutation group of degree n that is not A n or S n . (i) If G is semiprimitive, then either n > and d ( G ) = 2 , or n and d ( G ) . (ii) If G is soluble, then d ( G ) , and if G has odd order, then d ( G ) = 2 . (iii) If A s is not a section of G for some s > , then d ( G ) s . To finish this section, we discuss the distinguishing numbers of imprimitive wreath products.Chan proved in [18, Theorem 2.3] that for any H Sym(∆) and K S r , the distinguishingnumber of H ≀ K in its imprimitive action on ∆ × r is the minimal d for which H has at least d r ( K ) regular orbits on P d (∆). Since an orbit is regular whenever it consists of distinguishingpartitions, d > d ∆ ( H ). By Dolfi [21, Remark 2], if s > d ∆ ( H ), then H has at least s + 1 regularorbits on P s +1 (∆), so d max { d ∆ ( H ) + 1 , d r ( K ) } . Moreover, the following result, which wasused in [22] to prove Theorem 3.1, shows that d can be much smaller than d r ( K ); see [22,Lemma 2.4 and Remark 2.5] for a proof. Lemma 3.3.
Let G be a transitive permutation group on Ω . Let ∆ be a block of G with size m ,and let Σ := { ∆ g : g ∈ G } . Let H := G ∆∆ and K := G Σ . Then d Ω ( G ) d ∆ ( H ) ⌈ m p d Σ ( K ) ⌉ . Twisted wreath products
We begin by defining a twisted wreath product, as in [48]. Let T be a finite non-abelian simplegroup, and let P be a finite group acting transitively on k where k >
2. Let Q := P , the stabiliserin P of 1 ∈ k , and let ϕ : Q → Aut( T ) be a homomorphism such that core P ( ϕ − (Inn( T ))) = 1.For any subset X of P , let T X denote the set of functions from X to T , and note that T X formsa group under pointwise multiplication with identity e where e ( x ) := 1 for all x ∈ X . Let B := { f ∈ T P | f ( xq ) = f ( x ) ϕ ( q ) for all x ∈ P, q ∈ Q } . Then B T P , and P acts on B by f z ( x ) := f ( zx ) for z, x ∈ P and f ∈ B . Since ( f g ) z = f z g z for all z ∈ P and f, g ∈ B , we may define the twisted wreath product T twr ϕ P to be thesemidirect product B : P . Further, T twr ϕ P acts on B by f gx := ( f g ) x for f, g ∈ B and x ∈ P .Note that P = ( T twr ϕ P ) e . We refer to B as the base group of T twr ϕ P .Next we make some observations concerning the twisted wreath product G := T twr ϕ P andits action on B . Since P is transitive on k , we may choose a left transversal L := { a , . . . , a k } for Q in P such that i a i = 1 for all i ∈ k . Any element of T L can be extended to an element of B , so B = T × · · · × T k ≃ T k where T i := { f ∈ B : f ( a j ) = 1 for j ∈ k \ { i }} . Observe that if x ∈ P and i ∈ k , then q x,i := a − i xa i x ∈ Q , so for all f ∈ B ,(1) f x ( a i x ) = f ( a i ) ϕ ( q x,i ) . JOANNA B. FAWCETT
It follows that for all x ∈ P and i ∈ k ,(2) T xi = T i x . Hence B is a minimal normal subgroup of G . Since 1 = core P ( ϕ − (Inn( T ))) ≃ C G ( B ) by [4,Proposition 2.7], B = soc( G ). Further, 1 = core P (ker( ϕ )) = C P ( B ) by (1), so the action of G on B is faithful. Thus G is a quasiprimitive permutation group whose socle B is non-abelian,non-simple and regular. Further, P k is the top group of G by (2).We will use the following hypothesis to simplify notation for the remainder of the paper. Hypothesis 4.1.
Let T be a finite non-abelian simple group, and let P be a finite group actingtransitively on k where k >
2. Let Q := P , the stabiliser in P of 1 ∈ k , and let ϕ : Q → Aut( T )be a homomorphism such that core P ( V ) = 1 where V := ϕ − (Inn( T )). Let U := ker( ϕ ). Let G := T twr ϕ P , and let B be the base group of G . Let L := { a , . . . , a k } be a left transversal for Q in P such that i a i = 1 for all i ∈ k . For x ∈ P and i ∈ k , let q x,i := a − i xa i x ∈ Q .Next we have some basic but useful observations. Lemma 4.2.
Under Hypothesis , b B ( G ) = b B ( P ) + 1 .Proof. Since G is transitive on B , we may assume that a base of minimal size for G containsthe identity element e of B . Since P = G e , the result follows. (cid:3) Lemma 4.3.
Under Hypothesis , f ( a i ) ϕ ( q x,i ) = f ( a i x ) for all x ∈ P , i ∈ k and f ∈ fix B ( x ) .Proof. This is immediate from (1). (cid:3)
Recall from the introduction that a finite quasiprimitive permutation group G is of twistedwreath type whenever G has a unique non-abelian non-simple regular minimal normal subgroup.By the quasiprimitive version of the O’Nan-Scott theorem [48] (see § Theorem 4.4.
For a finite permutation group X , the following are equivalent. (i) X is quasiprimitive of twisted wreath type. (ii) X is permutation isomorphic to a twisted wreath product G acting on B where G and B are described by Hypothesis . Remark 4.5.
For any non-abelian simple group T and transitive permutation group P on k ,we can construct a quasiprimitive permutation group of twisted wreath type. Indeed, if Q isthe stabiliser in P of 1 ∈ k , then core P ( Q ) = 1, so we may choose ϕ : Q → Aut( T ) to be anyhomomorphism (such as the trivial one). There are also many examples where P is not faithfulon k . Let T := A n where n >
5, let P be any group of order 2 k with a central involution z , let Q := h z i , and define ϕ : Q → S n by z (12), so that core P ( ϕ − (Inn( T ))) = 1.Our next result is [4, Theorem 3.5], which characterises when a twisted wreath product isprimitive. Theorem 4.6.
Assume Hypothesis . Let M E P where M U = M V . Let U ′ := U ∩ M and V ′ := V ∩ M . Then G is primitive on B if and only if the following three conditions hold.(i) V ′ /U ′ ≃ T .(ii) Q = N P ( U ′ ) ∩ N P ( V ′ ) .(iii) If R is a subgroup of M normalised by Q for which R ∩ V ′ = U ′ , then R = U ′ . Note that we can always take the group M in Theorem 4.6 to be P itself, so this result doesindeed characterise when a twisted wreath product is primitive on its base group.One necessary condition of primitivity that is not immediate from Theorem 4.6 is that P actsfaithfully on k (i.e., core P ( Q ) = 1). This was first proved in [41, Case 2(a)]. We record thisresult below, along with a useful consequence of Theorem 4.6. ASES OF TWISTED WREATH PRODUCTS 7
Lemma 4.7.
If Hypothesis holds and G is primitive on B , then core P ( Q ) = 1 and Inn( T ) is a subgroup of ϕ ( Q ) .Proof. By [20, Theorem 4.7B], the action of P by conjugation on the k simple factors of B isfaithful, so P is faithful on k by (2). In other words, core P ( Q ) = 1. Taking M to be P inTheorem 4.6, we see that T ≃ V /U ≃ ϕ ( V ) Inn( T ), so Inn( T ) = ϕ ( V ) ϕ ( Q ). (cid:3) Primitive permutation groups of twisted wreath type are therefore quite rare. For example,Lemma 4.7 implies that T is a section of the symmetric group S k , so there are only finitely manyprimitive groups of twisted wreath type for each k .Assuming only the necessary conditions for primitivity that are given by Lemma 4.7, we nowdetermine the possibilities for T when P is S k or A k . Lemma 4.8.
Assume Hypothesis . If
Inn( T ) ϕ ( Q ) and P is S k or A k , then T ≃ A k − and k > .Proof. Since Q is S k − or A k − , the kernel of ϕ is trivial, A k − or S k − . The latter two casesare impossible, or else ϕ ( Q ) would be trivial or C , so ϕ is injective. Since Inn( T ) E ϕ ( Q ), itfollows that T ≃ A k − and k > (cid:3) Note that for any k >
6, there is a primitive permutation group of twisted wreath type withsocle A kk − and top group S k or A k (see Example 9.3).The following basic result provides a useful method for constructing examples of quasiprimitiveand primitive groups of twisted wreath type. Lemma 4.9.
Let H be a quasiprimitive permutation group on ∆ of twisted wreath type withsocle A ≃ T m where T is simple. Let Ω := ∆ r where r > . Let A r G H ≀ S r where theimage K of the projection map ρ : G → S r is transitive on r . Then G H ≀ K , and G is aquasiprimitive permutation group on Ω of twisted wreath type with socle A r . Further, if H isprimitive on Ω , then H ≀ K is primitive on Ω .Proof. The socle A of H is non-abelian, non-simple, regular and the unique minimal normalsubgroup of H . Thus B := A r is regular on Ω. Further, C H ( A ) = 1, so C G ( B ) = 1. Since H acts transitively by conjugation on the simple factors of A and K is transitive on r , it followsthat B is the unique minimal normal subgroup of G . Thus G is quasiprimitive of twisted wreathtype with socle B . If H is primitive, then H ≀ K is primitive by [20, Lemma 2.7A]. (cid:3) We caution the reader that although H ≀ K is primitive when H is primitive, there are groups G that satisfy the conditions of Lemma 4.9 but are not primitive.Recall the definition of a blow-up from §
2. As defined in [4], a primitive permutation groupof twisted wreath type is said to be of minimal-twisted type if it is not a blow-up of index r > § S of G is balanced if Q S P and core S ( U ) = core S ( V ).Let S ( G ) denote the set of balanced subgroups of G . Observe that S ( G ) is a partially orderedset under inclusion, and P ∈ S ( G ). Note that if G is primitive on B , then by Lemma 4.7, Q / ∈ S ( G ). The following result is a weaker version of [4, Theorems 8.3 and 9.12]. Theorem 4.10.
Assume Hypothesis where G is primitive on B , so core P ( Q ) = 1 . Thenone of the following holds. (i) S ( G ) = { P } , and one of the following holds. (a) P is almost simple and quasiprimitive on k . (b) P is primitive of diagonal type on k , and P has a unique minimal normal subgroupthat is isomorphic to T ℓ for some ℓ . JOANNA B. FAWCETT (ii) S ( G ) = { P } . Let S be minimal in S ( G ) . Let P := S/ core S ( Q ) , Q := Q/ core S ( Q ) and ϕ : Q → Aut( T ) be the homomorphism induced by ϕ . Then G := T twr ϕ P is primitiveon B := soc( G ) , and S ( G ) = { P } . Further, up to permutation isomorphism, B r G G ≀ S r , where r := [ P : S ] , the wreath product G ≀ S r acts via the product action on B r , and theimage of the projection map from G to S r is transitive on r .Proof. Suppose that S ( G ) = { P } . Then G is of minimal-twisted type by [4, Theorem 8.3(3)].By [4, Theorem 9.12], G is permutation isomorphic to a twisted wreath product G ′ = T ′ twr ϕ ′ P ′ described by [4, Constructions 9.1 or 9.3]. In particular, the point stabiliser Q ′ is core-free in P ′ ,so P ′ is permutation isomorphic to P . Thus either P is primitive of diagonal type with a uniqueminimal normal subgroup T ℓ for some ℓ , in which case (i)(b) holds, or P is almost simple, inwhich case P = soc( P ) Q by [4, Proposition 9.2(2)], so soc( P ) is transitive and (i)(a) holds.Thus we may assume that S ∈ S ( G ) is minimal, as in (ii). By [4, Proposition 8.4], N :=core S ( Q ) = core S ( V ) = core S ( U ). For any subset X of S that contains N , let X := X/N ,and let P := S , as in (ii). Now core P ( Q ) = 1, so we may define G as in (ii). Note that V = ϕ − (Inn( T )) and U = ker( ϕ ). If R ∈ S ( G ), then Q R S and core R ( U ) = core R ( V ), so R ∈ S ( G ) and R S . Thus R = S , so S ( G ) = { P } .Since S contains the stabiliser Q = P , it follows that S = P I for some subset I of k with1 ∈ I . In particular, the set A := { a i : i ∈ I } is a left transversal for Q in S with the propertythat i a i = 1 for i ∈ I (see Hypothesis 4.1). Further, A is a left transversal for Q in P . Thus B ≃ T A ≃ T A ≃ { f ∈ B : f ( x ) = 1 for x ∈ P \ S } =: B ( S ) . Note that C S ( B ( S )) = N by (1), and G B ( S ) = B ( S ): S . It is then routine to verify that G = B : P is permutation isomorphic to G B ( S ) /N in its action on B ( S ). By the proof of [4, Theorem 8.3(2)], G is a blow-up of G B ( S ) /N , so G is a blow-up of G . Thus G is primitive, and (ii) holds. (cid:3) Remark 4.11.
We make several observations concerning Theorem 4.10.(1) If the top group of a primitive group of twisted wreath type is primitive, then certainly G is of minimal-twisted type, but the converse is not true (see [4, Example 9.8]).(2) The groups P described in (i)(b) are precisely the primitive groups of type III(a)(i)in Table 1. In fact, a primitive twisted wreath product can be constructed using anyquasiprimitive P in class III(a)(i) in a prescribed way (see Example 9.1). Thus theminimal-twisted type groups whose top groups are primitive of diagonal type are wellunderstood. We also conclude that there are primitive twisted wreath products that arenot of minimal-twisted type but still have quasiprimitive top groups.(3) Much less is known about the almost simple groups P described in (i)(a); see [4, Con-struction 9.1 and § T , there is a minimal-twisted type group with socle T k for some k whose top group isalmost simple and primitive. See [4, §
9] for more examples.(4) The set S ( G ) may contain more than one minimal element; see [4, Example 9.11].5. Probabilistic results
In this section, we prove the following result, and then we use this result to prove Theorem 1.5.Some of the methods in this section will also be used to prove Theorem 1.1.
Theorem 5.1.
Suppose that Hypothesis holds and core P ( Q ) = 1 . If P is primitive on k ,and if Inn( T ) ϕ ( Q ) when P is S k or A k , then the proportion of pairs of points from B thatare bases for G tends to 1 as | G | → ∞ . ASES OF TWISTED WREATH PRODUCTS 9
First we need some notation. For a transitive permutation group G on Ω and an integer b >
1, let Q ( G, b ) denote the proportion of b -tuples in Ω b that are not (ordered) bases for G .The following observation was made by Liebeck and Shalev [43, p.502]. Lemma 5.2.
Let G be a transitive permutation group on Ω . For any integer b > and α ∈ Ω , Q ( G, b ) X x ∈ R | x G ∩ G α | b | C G ( x ) | b − | G | b − , where R is any set of representatives for the G -conjugacy classes of prime order elements in G α . In order to apply Lemma 5.2 in the case where G is a twisted wreath product, we require twobasic results concerning conjugacy classes and centralisers of elements of G . Lemma 5.3.
Under Hypothesis , the following hold for x ∈ P . (i) x G ∩ P = x P . (ii) C G ( x ) = { f y ∈ G : f ∈ fix B ( x ) , y ∈ C P ( x ) } . (iii) | C G ( x ) | = | fix B ( x ) || C P ( x ) | .Proof. (i) Let y ∈ x G ∩ P . Then y = ( f z ) − x ( f z ) for some f ∈ B and z ∈ P . Now ( f z ) − x ( f z ) =( f − f x − ) z ( z − xz ). Since G = B : P and y ∈ P , we must have y = z − xz ∈ x P .(ii) Let f ∈ B and y ∈ P . Then f y centralises x if and only if f ( yx ) = f x − ( xy ), and thisoccurs precisely when f ∈ fix B ( x ) and y ∈ C P ( x ).(iii) This follows from (ii) since G = B : P . (cid:3) Lemma 5.4.
Under Hypothesis , | fix B ( x ) | | T | ω k ( x ) for all x ∈ P .Proof. Let x ∈ P . Without loss of generality, we may assume that the r := ω k ( x ) orbits of h x i on k have representatives 1 , . . . , r . Recall that { a , . . . , a k } is a left transversal for Q in P .Define a map ψ : fix B ( x ) → T r by f ( f ( a ) , . . . , f ( a r )). We claim that ψ is injective. To thisend, suppose that ( f ( a ) , . . . , f ( a r )) = ( f ( a ) , . . . , f ( a r )) for some f , f ∈ fix B ( x ). Let i ∈ k .Now i = j y for some j ∈ r and y ∈ h x i . By Lemma 4.3, f ( a j ) ϕ ( q y,j ) = f ( a j y ) for f ∈ { f , f } , so f ( a i ) = f ( a j y ) = f ( a j ) ϕ ( q y,j ) = f ( a j ) ϕ ( q y,j ) = f ( a j y ) = f ( a i ) . Thus f = f , and the claim holds. (cid:3) Next we obtain a useful version of Lemma 5.2 for twisted wreath products.
Lemma 5.5.
Assume Hypothesis . For any integer b > , Q ( G, b ) X x ∈ R ( P ) | x P || T | ( b − k − ω k ( x )) , where R ( P ) is any set of representatives for the conjugacy classes of prime order elements in P .Proof. By Lemma 5.3(i), R ( P ) is a set of representatives for the G -conjugacy classes of primeorder elements in G e = P , and | x G ∩ G e | = | x P | for all x ∈ R ( P ). Moreover, Lemma 5.3(iii) andLemma 5.4 imply that | C G ( x ) | = | fix B ( x ) || C P ( x ) | | T | ω k ( x ) | C P ( x ) | for all x ∈ R ( P ). Since | G | = | T | k | P | , Lemma 5.2 implies that Q ( G, b ) X x ∈ R ( P ) | x P | b ( | T | ω k ( x ) | C P ( x ) | ) b − ( | T | k | x P || C P ( x ) | ) b − , and the result follows. (cid:3) Proof of Theorem . By assumption, P is a primitive subgroup of S k , and Inn( T ) ϕ ( Q )when P is S k or A k . It suffices to prove that Q ( G, → | G | → ∞ . Let R ( P ) be a set of representatives for the conjugacy classes of prime order elements in P . By Lemma 5.5, Q ( G, P x ∈ R ( P ) | x P | / | T | k − ω k ( x ) . We claim that X x ∈ R ( P ) | x P || T | k − ω k ( x ) C | T | c k + 1 √ k + k + 4 k ( k − ! for some absolute constants C and c >
1, in which case Q ( G, → | G | → ∞ since the factthat | G | | T | k k ! forces | T | → ∞ or k → ∞ as | G | → ∞ .If P is not S k or A k , then the claim holds by [24, Lemma 4.4], so we assume that P is S k or A k . Then T = A k − by Lemma 4.8. Note that k − ω k ( x ) is 1 when x is a transposition and atleast 2 otherwise. Then X x ∈ R ( P ) | x P || T | k − ω k ( x ) − | (12) S k || T | − + | S k || T | − = k ( k − k − + 2 k ( k − k + 4 k ( k − , and the claim follows. (cid:3) Remark 5.6.
It would be interesting to determine whether Theorem 5.1 still holds if we weakenthe assumption that P is primitive to quasiprimitive. By Theorem 7.1, such twisted wreathproducts have base size 2, so this is a reasonable problem to consider. It would suffice to adaptthe proof of [24, Lemma 4.4]. The key ingredients of this proof are the following properties ofprimitive subgroups P of S k : (a) if P contains a transposition, then P = S k ; (b) either A k P ,or | P | k [49]; (c) either A k P , or | P | exp(4 √ k log k ) for sufficiently large k [3]; (d)either the minimal degree of P is at least k/
3, or A rm P S m ≀ S r and P acts via the productaction on ∆ r where ∆ is the set of ℓ -subsets of m and m > r > ℓ < m/ P of S k . Proof of Theorem . Let G be a primitive permutation group of twisted wreath type whosetop group is primitive. By Theorem 4.4, we may assume that G is given by Hypothesis 4.1, so G is primitive on B and the top group P k is primitive on k . By Lemma 4.7, core P ( Q ) = 1 andInn( T ) ϕ ( Q ), so the result follows from Theorem 5.1. (cid:3) Bounds on base sizes
In this section, we establish two upper bounds on the base size of a twisted wreath product.The proof of the first bound is elementary, but we use the CFSG in the second to deduce that T is 2-generated. In the following, recall the definition of the function ω from § Lemma 6.1.
If Hypothesis holds and core P ( Q ) = 1 , then b B ( G ) (cid:24) log d k ( P )log ω T ( ϕ ( Q )) (cid:25) + 1 . Proof.
Let d := d k ( P ), n := ω T ( ϕ ( Q )) and m := ⌈ log d/ log n ⌉ . Note that d >
2, so m >
1. Notealso that n m − d − < n m . For each integer u such that 0 u d −
1, let d u, , . . . , d u,m − denote the first m digits of the base n representation of u , so u = d u, + d u, n + · · · + d u,m − n m − . Let t , . . . , t n − be representatives of the orbits of ϕ ( Q ) on T , and let { ∆ , . . . , ∆ d − } be adistinguishing partition for P on k . Recall from Hypothesis 4.1 that L = { a , . . . , a k } is a lefttransversal for Q in P , and recall from § B , it sufficesto define an element of T L . For 1 j m , define f j ( a i ) := t d u,j − whenever i ∈ ∆ u . Then B := { f , . . . , f m } ⊆ B . Suppose that x ∈ P fixes B pointwise. Let i ∈ ∆ u where 0 u d − i x ∈ ∆ v for some 0 v d −
1. By Lemma 4.3, f j ( a i ) ϕ ( q x,i ) = f j ( a i x ) for 1 j m ,so t d u,j − and t d v,j − are in the same orbit of ϕ ( Q ) for 1 j m . Thus d u,j − = d v,j − for ASES OF TWISTED WREATH PRODUCTS 11 j m , so u = v . We have proved that ∆ xu = ∆ u for 0 u d −
1, so x ∈ core P ( Q ) = 1.Hence B is a base for the action of P on B , and we are done by Lemma 4.2. (cid:3) Lemma 6.2.
If Hypothesis holds, then b B ( G ) (cid:24) log d k ( P )log | T | (cid:25) + 3 . Proof.
Let d := d k ( P ), n := | T | and m := ⌈ log d/ log n ⌉ . For each integer u such that 0 u d −
1, let d u, , . . . , d u,m − denote the first m digits of the base n representation of u , so u = d u, + d u, n + · · · + d u,m − n m − . By [1], there exist t, s ∈ T such that T = h t, s i . Enumerate the elements of T as t , . . . , t | T |− , andlet { ∆ , . . . , ∆ d − } be a distinguishing partition for P on k . For 1 j m , let f j ( a i ) := t d u,j − whenever i ∈ ∆ u . Let f m +1 ( a i ) := t and f m +2 ( a i ) := s for i ∈ k . Then B := { f , . . . , f m +2 } ⊆ B .Suppose that x ∈ P fixes B pointwise. By Lemma 4.3, f ( a i ) ϕ ( q x,i ) = f ( a i x ) for all i ∈ k and f ∈ B . In particular, for each i ∈ k , the automorphism ϕ ( q x,i ) fixes t and s , so ϕ ( q x,i ) = 1. Let i ∈ ∆ u where 0 u d −
1. Then i x ∈ ∆ v for some 0 v d −
1. Now t d u,j − = f j ( a i ) = f j ( a i x ) = t d v,j − for 1 j m . Thus d u,j − = d v,j − for 1 j m , so u = v . We have provedthat ∆ xu = ∆ u for 0 u d −
1, so x ∈ core P ( Q ). For i ∈ k , note that q x,i ∈ U since ϕ ( q x,i ) = 1,and observe that q x,i = a − i xa i x = a − i xa i . Thus x ∈ T i ∈ k a i U a − i = core P ( U ) core P ( V ) = 1.Hence B is a base for P , and we are done by Lemma 4.2. (cid:3) To finish this section, we use Theorem 3.1 to show that there is a close relationship betweenthe upper bound on b B ( G ) in Lemma 6.2 and the trivial lower bound log | B | | G | b B ( G ). Lemma 6.3.
Assume Hypothesis . Then (3) log d k ( P )log | T | < log | G | log | B | . Further, if core P ( Q ) = 1 , then (4) log | G | log | B | < log d k ( P )log | T | + 1 . Proof.
Let t := | T | . Recall that | G | = | B || P | and | B | = t k . Let R := P k , so d := d k ( P ) = d k ( R ).By Theorem 3.1, d k p | R | k p | P | , and clearly 48 < t , solog t d log t (48 k p | P | ) < t k p | P | = 1 + log t k | P | = log | B | | G | , in which case (3) holds. Now suppose that core P ( Q ) = 1. Then k p | P | < d (see § (cid:3) Base size § Theorem 7.1.
Suppose that Hypothesis holds and core P ( Q ) = 1 . If any of the followinghold, then b B ( G ) = 2 . (i) P is semiprimitive on k , and if P is S k or A k , then Inn( T ) ϕ ( Q ) . (ii) A is not a section of P . (iii) The order of P is not divisible by . (iv) A is not a section of P and T A . (v) P is soluble and T A . (vi) A s is not a section of P and s ω T (Aut( T )) . Remark 7.2.
We make several observations concerning Theorem 7.1.(1) For any T and P that satisfy one of Theorem 7.1(i)–(vi), there is a quasiprimitive groupof twisted wreath type with socle T k and top group P by Remark 4.5.(2) If Hypothesis 4.1 holds and G is primitive on B , then T is a section of P by Lemma 4.7,so (iii) and (v) cannot hold. Moreover, if (i) holds, then P is quasiprimitive. Indeed, anysemiregular normal subgroup of P is trivial, for if S is such a subgroup, then S ∩ Q = 1,so ( SU ) ∩ V = U , but then we may take M = P and R = SU in Theorem 4.6(iii), so R = U and S = 1. (The author thanks Luke Morgan for providing this argument.)(3) In Example 9.2, for any non-abelian simple group T , we construct an infinite family ofprimitive twisted wreath products T twr ϕ P with base size 2 where P is not quasiprimi-tive. The problem of classifying the primitive groups of twisted wreath type with basesize 2 therefore remains open. See also Example 9.6.(4) The assumption that Inn( T ) ϕ ( Q ) when P is S k or A k is required; see Example 9.5.In order to prove Theorem 7.1, we first consider the case where P is S k or A k . Lemma 7.3.
Assume Hypothesis . If
Inn( T ) ϕ ( Q ) and P is S k or A k , then b B ( G ) = 2 .Proof. By Lemma 4.8, T = A k − and k >
6. First suppose that k >
8. Let m := k −
1. Now d k ( P ) m + 1, so by Lemma 6.1, it suffices to show that m < ω A m ( S m ). Observe that ω A m ( S m )is the number of cycle types of even permutations of m . Write m + 1 = 2 i r where i is a non-negative integer and r is an odd integer. Then A m +1 contains a fixed-point-free permutationconsisting of 2 i disjoint r -cycles unless r = 1, in which case A m +1 contains a fixed-point-freepermutation consisting of 2 i − disjoint transpositions. Thus ω A m ( S m )+1 ω A m +1 ( S m +1 ). Since ω A ( S ) = 8, the result follows by induction.It remains to consider the case where k = 6 or 7. Let R ( P ) be a set of representatives forthe conjugacy classes of prime order elements in P . By Lemma 5.5, it suffices to prove that P x ∈ R ( P ) | x P || T | ω k ( x ) < | T | k , for then Q ( G, <
1, so b B ( G ) = 2. This is routine. (cid:3) Proof of Theorem . By Lemma 6.1, b B ( G ) = 2 if d k ( P ) ω T (Aut( T )). By Theorem 3.2(iii),if A s is not a section of P , then d k ( P ) s . Hence if (vi) holds, then b B ( G ) = 2.Note that ω T (Aut( T )) > | T | by Burnside’s p a q b Theorem [34,Theorem 31.4]. Thus b B ( G ) = 2 if d k ( P )
4. In particular, if (ii) or (iii) holds, then (ii) holds,so b B ( G ) = 2. Moreover, if (i) holds, then b B ( G ) = 2 by Theorem 3.2(i) and Lemma 7.3.By [55, Theorem 2.3], ω T (Aut( T )) = 4 if and only if T ≃ A . Thus b B ( G ) = 2 if d k ( P ) T A . In particular, if (iv) or (v) holds, then (iv) holds, so b B ( G ) = 2. (cid:3) Proof of Theorem . By assumption, G is a primitive permutation group of twisted wreathtype whose top group is quasiprimitive. By Theorem 4.4, we may assume that G is given byHypothesis 4.1, so G is primitive on B , and P k is quasiprimitive. By Lemma 4.7, core P ( Q ) =1 and Inn( T ) ϕ ( Q ), so the result follows from Theorem 7.1(i) (since any quasiprimitivepermutation group is semiprimitive by definition). (cid:3) Results related to Pyber’s conjecture
In this section, we first prove the following result, which determines the base sizes of a largeclass of quasiprimitive groups of twisted wreath type up to three possible values (and sometimestwo values). We then use this result to prove Theorems 1.3 and 1.4.
Theorem 8.1.
Let H be a quasiprimitive permutation group on ∆ of twisted wreath type withsocle A ≃ T m where T is simple. Let Ω := ∆ r where r > . Let A r G H ≀ S r where theimage K of the projection map ρ : G → S r is transitive on r . Then G H ≀ K , and G is aquasiprimitive permutation group on Ω of twisted wreath type. If b ∆ ( H ) = 2 , then b Ω ( G ) = (cid:24) log | G | log | Ω | (cid:25) + ε = (cid:24) log d r ( K ) m log | T | (cid:25) + δ ASES OF TWISTED WREATH PRODUCTS 13 for some ε, δ ∈ { , , } where ε δ . Further, if G = H ≀ K and [ H : A ] > , then ε ∈ { , } , δ ∈ { , } and ε + 1 δ . To prove Theorem 8.1, we require the following elementary result, which we prove by adaptingthe proof of Lemma 6.2.
Lemma 8.2.
Let H Sym(∆) and K S r where | ∆ | > and K = 1 . Let G H ≀ K and Ω := ∆ r . Then b Ω ( G ) (cid:24) log d r ( K )log | ∆ | (cid:25) + b ∆ ( H ) . Proof.
Let d := d r ( K ), n := | ∆ | and m := ⌈ log d/ log n ⌉ . Note that n m − d − < n m . We mayassume that ∆ = { , . . . , n − } . For each integer u such that 0 u d −
1, let d u, , . . . , d u,m − denote the first m digits of the base n representation of u , so u = d u, + d u, n + · · · + d u,m − n m − and d u, , . . . , d u,m − ∈ ∆. Let { ∆ , . . . , ∆ d − } be a distinguishing partition for K on r . For1 j m , let α j be the element of ∆ r whose i -th coordinate is d u,j − whenever i ∈ ∆ u . Let b := b ∆ ( H ), and let i , . . . , i b be a base for H on ∆. For 1 ℓ b , let β ℓ := ( i ℓ , . . . , i ℓ ) ∈ ∆ r .Let B := { α , . . . , α m , β , . . . , β b } and suppose that g = ( h , . . . , h r ) π ∈ G fixes B pointwise.Then h j fixes i ℓ for 1 ℓ b and 1 j r , so g = (1 , . . . , π . Let i ∈ ∆ u . Now i π ∈ ∆ v for some v . For 1 j m , the digit d u,j − is the i -th coordinate of α j , so d u,j − is the i π -thcoordinate of α πj = α j , so d u,j − = d v,j − . Thus u = v . We have shown that ∆ πu = ∆ u for0 u d −
1, so π = 1. Hence B is a base for G , and the bound on b Ω ( G ) follows. (cid:3) Note that the bound of Lemma 8.2 is stated in [32, § H is primitiveand almost simple and K is transitive. They use [22, Lemma 2.1] to deduce that m is the basesize of K in its action on the set of partitions of r with at most | ∆ | parts, and there is a naturalway of using such a base to construct α , . . . , α m so that the proof proceeds as above (see theproof of [8, Lemma 3.8]). Thus our proof is essentially the same but more constructive. Proof of Theorem . By Lemma 4.9, G is a quasiprimitive permutation group on Ω of twistedwreath type with socle B := A r . Note that | ∆ | = | A | = | T | m and | Ω | = | B | = | T | mr .Let c := 1 when G = H ≀ K and [ H : A ] >
48, and let c := 0 otherwise. Note that | B || K | | G | since B ker( ρ ), and if c = 1, then 48 r | B || K | | G | . By Theorem 3.1,log d r ( K ) m log | T | + c log(48 r p | K | ) m log | T | + c = log 48 m log | T | + log( | B || K | )log | B | + c − log | G | log | Ω | . Since log | Ω | | G | b Ω ( G ) and b ∆ ( H ) = 2, the result follows from Lemma 8.2. (cid:3) Proof of Theorem . Recall that G is a quasiprimitive permutation group on Ω of twistedwreath type with top group R . By Theorem 4.4, we may assume that G is given by Hypothe-sis 4.1, in which case Ω = B , and we may assume that R = P k , so d := d k ( R ) = d k ( P ). Recallthat log | Ω | | G | b Ω ( G ). By Lemmas 6.2 and 6.3(3), b Ω ( G ) = ⌈ log | Ω | | G |⌉ + ε = ⌈ log | T | d ⌉ + δ forsome ε, δ ∈ { , , , } where ε δ . In particular, the first claim in Theorem 1.3 holds.Now suppose that G is primitive on Ω. Then core P ( Q ) = 1 by Lemma 4.7, so δ ε + 1 byLemma 6.3(4). It remains to prove that ε
2. If P is quasiprimitive on k , then b Ω ( G ) = 2 byTheorem 1.1, so ε
2. Thus we may assume that P is not quasiprimitive, so Theorem 4.10(ii)holds for G . Let H := G and ∆ := B as in Theorem 4.10(ii). By Theorem 4.10, H is primitive oftwisted wreath type with a quasiprimitive top group, so b ∆ ( H ) = 2 by Theorem 1.1. Moreover,by Theorem 4.10(ii), G and H satisfy the conditions of Theorem 8.1, so ε
2, as desired. (cid:3)
Proof of Theorem . Recall that G = H ≀ K where H is a primitive permutation group on∆ of twisted wreath type with socle T m , and K is a transitive subgroup of S r where r > G is a primitive permutation group of twisted wreath type on Ω = ∆ r . By Theorem 4.4 and Lemma 4.7, T is a section of the top group R of H , and [ H : T m ] = | R | , so[ H : T m ] > | T | >
48. By assumption, R is quasiprimitive, so b ∆ ( H ) = 2 by Theorem 1.1. Thusthe result follows from Theorem 8.1. (cid:3) Examples
The following is [4, Example 4.8], which demonstrates (in particular) that for any finite non-abelian simple group T , there are primitive permutation groups of twisted wreath type withsocle T k for infinitely many k , and for all of these groups, the top group is quasiprimitive. Example 9.1.
Let T be a finite non-abelian simple group, and let ℓ >
2. Let P be a quasiprim-itive permutation group of diagonal type with socle A := T ℓ , where A is the unique minimalnormal subgroup of P . (These are precisely the groups of type III(a)(i) in Table 1.) We mayview P as a subgroup of S k where k := | T | ℓ − , in which case Q := P is isomorphic to a subgroupof Aut( T ) × S k , and T ≃ A ∩ Q E Q . For each x ∈ Q , we may define ϕ ( x ) ∈ Aut( A ∩ Q ) tobe conjugation by x , and since core P ( Q ) = 1, it follows that G := T twr ϕ P is a quasiprimitivepermutation group. In fact, G is primitive by Theorem 4.6 (see [4, Lemma 4.4] for details). Notethat P is primitive on k if and only if the permutation group P ℓ induced by the action of P onthe ℓ simple factors of A is primitive. Also, for any transitive R S ℓ , we can choose P so that P ℓ = R . Thus the top group of G can be either primitive, or quasiprimitive and not primitive.Using Example 9.1, we construct a large family of primitive permutation groups of twistedwreath type with base size 2 whose top groups are not quasiprimitive. Example 9.2.
Let T be any finite non-abelian simple group, and let ℓ >
2. Let m := | T | ℓ − .By Example 9.1, there is a primitive permutation group H of twisted wreath type with socle T m and top group S where S is quasiprimitive on m with socle T ℓ . Let K be any transitivepermutation group of degree r > r m , or K is soluble or semiprimitive(but not S r or A r ). Then G := H ≀ K is a primitive permutation group of twisted wreath typewith socle T mr by Lemma 4.9. The top group of G is P := S ≀ K in its imprimitive action on m × r , and P is not quasiprimitive since S r fixes the r blocks of imprimitivity. By Theorem 3.2, d m ( S ) = 2 and d r ( K ) m , so by Lemma 3.3, d m × r ( P )
4. As we saw in the proof ofTheorem 7.1, ω T (Aut( T )) >
4. Thus b ( G ) = 2 by Lemma 6.1.Next we show that for any finite non-abelian simple group T , there is a primitive permutationgroup G of twisted wreath type with socle T m for some m such that the top group P of G isprimitive and almost simple. In Example 9.3, we construct G for an almost simple group P withcertain properties, and then in Lemma 9.4, we show that some P with these properties can beconstructed for every T . Example 9.3.
Let P be an almost simple primitive permutation group on k such that Q := P is almost simple with socle T . For each x ∈ Q , define ϕ ( x ) ∈ Aut( T ) to be conjugation by x . Since core P ( Q ) = 1, it follows that G := T twr ϕ P is a quasiprimitive permutation group.To show that G is primitive, it suffices to show that conditions (i)–(iii) of Theorem 4.6 holdwith M = P . Let U := ker( ϕ ) = 1 and V := ϕ − (Inn( T )) = T . Clearly (i) holds. Since P is a primitive permutation group, Q = N P ( T ), so (ii) holds. For (iii), let R be a non-trivialsubgroup of P that is normalised by Q for which R ∩ T = 1. Then R ∩ Q = 1 and R E P , so R is a non-trivial regular normal subgroup of P , contradicting the O’Nan-Scott theorem. Thus R = 1 and (iii) holds, so G is primitive by Theorem 4.6. To see that such a group P exists forany non-abelian simple group T , see Lemma 9.4. Lemma 9.4.
Let T be a non-abelian simple group. Then there is an almost simple primitivepermutation group whose point stabiliser is almost simple with socle T .Proof. We claim that there is an almost simple group G with socle T and an almost simplegroup H with socle S such that G is maximal in H and S = T . If so, then G is core-free in H , ASES OF TWISTED WREATH PRODUCTS 15 so H is a primitive permutation group on the set of cosets of G , and we are done. If T = A m ,then we may take ( G, H ) = ( S m , S m +1 ), so we may assume that T A m for m >
5. Choose amaximal subgroup M of T , and let n := [ T : M ]. Then we may view T as a primitive subgroupof S n , and T is not regular. Let N := N S n ( T ). Then N is almost simple with socle T , and N A n has socle A n . If N is maximal in N A n , then we may take ( G, H ) = (
N, N A n ). Otherwise, thereis a group K such that N < K < N A n and N is maximal in K . If K is almost simple withsocle S , then S = T , so we may take ( G, H ) = (
N, K ). Otherwise, K is a primitive subgroup of S n that is not almost simple. By [40], T is PSL (7), M or Sp ( q ) where q is even. By [40], wemay take ( G, H ) to be (PSL (7):2 , S ), ( M , A ) or (Sp ( q ) , SL ( q )), respectively. (cid:3) We finish this section with two more examples.
Example 9.5.
In Theorem 7.1, we proved that if Hypothesis 4.1 holds where P is S k or A k and Inn( T ) ϕ ( Q ), then b B ( G ) = 2. The assumption that Inn( T ) ϕ ( Q ) cannot be removed.For example, by Remark 4.5, quasiprimitive permutation groups G of twisted wreath type withsocle A k and top group S k exist for all k >
2, but b B ( G ) > log k by Theorem 1.3. Example 9.6.
Assume Hypothesis 4.1 where G is primitive on B . Let K be a transitivesubgroup of S r where r >
2. By Lemma 4.9, G ≀ K is a primitive permutation group of twistedwreath type on B r . Moreover, the top group of G ≀ K is P ≀ K in its imprimitive action on k × r .In particular, by Example 9.3, we may take T = A k − , P = S k and K = S r where k >
6, inwhich case G k,r := G ≀ K has top group S k ≀ S r . By Theorem 1.4, b ( G k,r ) = (cid:24) log rk log | A k − | (cid:25) + δ k,r for some δ k,r ∈ { , } . In particular, b ( G k,r ) can be arbitrarily large, but also very small. Forexample, b ( G k,r ) ∈ { , } for all r (( k − / k . In fact, b ( G k,k ) = 2 for k > d k × k ( S k ≀ S k ) = k + 1, and by the proof of Lemma 7.3, k + 1 ω A k − ( S k − ) for k > X is a primitive group of twisted wreath type with topgroup S k ≀ S r acting on k × r , then X is permutation isomorphic to G k,r . References [1]
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