Groups with bounded centralizer chains and the Borovik–Khukhro conjecture

Introduction

Following [10, 24], the maximal length of nested chains of centralizers of subsets in a group G is called the c-dimension of G (the same number is also known as the length of the centralizer lattice of G, see, e.g., [25]). Here we consider groups of finite c-dimension. The class of such groups is quite wide: it includes, for example, abelian groups, linear groups, torsion-free hyperbolic groups, in particular, free groups. It is closed under taking subgroups, finite direct products, and extensions by finite groups, and is not closed under taking homomorphic images and arbitrary extensions. Obviously, every group of finite c-dimension satisfies the minimal condition on centralizers, therefore, it is a so-called 𝔐C-group.

The minimal condition on centralizers is a useful notion in the class of locally finite groups. Indeed, locally finite 𝔐C-groups satisfy an analogue of the Sylow theorem [4, Theorem B]. Periodic locally soluble 𝔐C-groups satisfy an analogue of the Hall theorem on π-subgroups [5, Theorem 1.6]; moreover, these groups are soluble and contain a nilpotent-by-abelian subgroup of finite index [5, Theorems 2.1 and 2.2]. When we restrict ourselves to the class of locally finite groups of finite c-dimension, it is natural to ask if the aforementioned results can be strengthened. For example, is the derived length of a locally soluble group in this class bounded in terms of the c-dimension? Another natural question is whether or not such a group contains a nilpotent-by-abelian subgroup of finite index bounded in the same terms. The first question has been answered in the affirmative [21, Theorem 1]. The answer to the second question is negative [21, Example 1], but a weaker statement holds: if G is a periodic locally soluble group of c-dimension k, then the quotient group of G by the second Hirsch–Plotkin radical F2⁢(G) contains an abelian subgroup of k-bounded index.

In the same paper [21], E. I. Khukhro posed the following conjecture which he attributed to A.V. Borovik.

Recall that the Hirsch–Plotkin radical F⁢(G) is the maximal normal locally nilpotent subgroup. The i-th Hirsch–Plotkin radical Fi⁢(G) is defined inductively: F1⁢(G)=F⁢(G) and Fi⁢(G) is the full inverse image of F⁢(G/Fi-1⁢(G)) in G. The layer E⁢(G) of G is the subgroup generated by all components of G, that is, its subnormal quasisimple subgroups. The generalized Fitting subgroup F*⁢(G) is the product of F⁢(G) and E⁢(G).

While part (1) of the conjecture is known to be true [7], the second part turns out to be false.

It is clear that a counterexample proving Theorem 1 should be an infinite series of locally finite groups G of uniformly bounded c-dimensions with unbounded minimal indices of abelian subgroups in the quotients G/S. In fact, we construct infinitely many counterexamples as follows. For every integer n>0, we define an infinite series 𝒢n of finite groups Gn,r, where r runs over an infinite set of primes. The c-dimensions of groups in 𝒢n are bounded in terms of n only. The quotient of G=Gn,r by the corresponding subgroup S (that is, the full inverse image of the generalized Fitting subgroup of Gn,r/F⁢(Gn,r)) is isomorphic to the symplectic group Sp2⁢n⁡(r). This contradicts statement (2) of the Borovik–Khukhro conjecture, because the minimum of the indices of the proper subgroups of G/S=Sp2⁢n⁡(r) increases with r.

It is worth noting that our construction depends on some deep number-theoretical results. It requires some results on Dickson’s conjecture and Schinzel’s hypothesis H on prime values of finite sets of linear forms and irreducible integer-valued polynomials obtained in [1]. We are thankful to participants of the forum on mathoverflow.net [28], especially to T. Tao, and to J. Maynard for the information on various results on these conjectures and related sieve methods. There is a more detailed discussion on this topic after the proof of Theorem 1 in Section 2.

Though the second part of the Borovik–Khukhro conjecture is false, a weaker version of it holds, at least in the case of finite groups.

The proof of Theorem 2 relies on [21, Theorem 1] and ideas from [7]. The difference from the proof in [7] is that we bound not just the number of nonabelian composition factors but the overall “size” of them.

Theorem 2 implies that, for a finite group G, the quotient of G/F3⁢(G) by the generalized Fitting subgroup contains an abelian subgroup of index bounded in terms of the c-dimension, while the conjecture was that F⁢(G) instead of F3⁢(G) is enough. So the natural question is whether or not one can replace F3⁢(G) by F2⁢(G). Furthermore, Theorem 2 considers only finite groups. Hence it would be interesting to obtain some positive statement for locally finite groups. Since every iterated Hirsch–Plotkin radical is contained in the locally soluble radical, the following seems to be the “weakest” analogue of the conjecture: Let G be a locally finite group of finite c-dimension k, and R the (locally) soluble radical of G; the quotient group of G/R by the generalized Fitting subgroup contains an abelian subgroup of finite index bounded in terms of k. The proof of this statement for finite groups follows from Theorem 2 (see Corollary 2 in Section 3).

Section 1 contains preliminary results on centralizers and some information on simple groups. In Section 2, we give a counterexample to the Borovik–Khukhro conjecture and discuss for which c-dimensions such counterexamples exist. Theorem 2 is proved in Section 3.

1 Preliminaries

The following lemma lists some well-known properties of centralizers.

Let G be a group and V a normal abelian subgroup of G. Let ¯:G→G/V be the natural homomorphism. The conjugation action of G on V induces an action of G/V on V. In the following lemma the centralizer CV⁢(Y¯) for Y⩽G is defined with respect to this action, and therefore coincides with CV⁢(Y).

The following lemma is crucial to the proof of Theorem 2.

For a finite group G, we define a nonnegative integer λ⁢(G) as follows. First, let G be a nonabelian simple group. If G is a group of Lie type, then λ⁢(G) is the minimal Lie rank of groups of Lie type isomorphic to G (the Lie rank is the rank of the corresponding (B,N)-pair [3, p. 249]). If G is an alternating group, then λ⁢(G) is the degree of G (except for the groups Alt5, Alt6 and Alt8 which are isomorphic to groups of Lie type and, therefore, already have assigned values of λ⁢(G)). Put λ⁢(G)=1 for the sporadic groups. If G is an arbitrary finite group, then λ⁢(G) is the sum of λ⁢(S), where S runs over the nonabelian composition factors of G.

Recall that the p-rank of a group G for a given prime p is the largest rank of elementary abelian p-subgroups of G.

The next lemma directly follows from the formulas for the orders of groups of Lie type [3, Table 16.1] and the well-known Zsigmondy theorem [27].

Denote by μ⁢(G) the minimal degree of a faithful permutation representation of a finite group G.

Note that Lemma 1.9 can also be deduced from the information on the values of μ⁢(S) for finite simple groups S of Lie type which are all known (the complete list of these numbers can be found, for example, in [17, Table 4]).

The following lemma is a direct consequence of [21, Theorem 1].

2 Counterexamples to the Borovik–Khukhro conjecture

Denote by Ω⁢(n) the number of prime divisors of a positive integer n counting multiplicities, i.e. if n=p1α1⁢⋯⁢pmαm for primes p1,…,pm, then

For positive integers n and M, we set

It is clear that the following inclusions hold for every n:

A crucial ingredient in constructing our counterexamples is the following number-theoretical statement.

Given n, put

Thus πn is the first infinite set in the chain (2.1).

Fix a positive integer n and an odd prime r. Let Rn,r be an extra-special group of order r2⁢n+1 and exponent r. It is known that Aut⁡(Rn,r) is split over Inn⁡(Rn,r) (see [16, p. 404]) and the image in Out⁡(Rn,r) of the centralizer in Aut⁡(Rn,r) of Z⁢(Rn,r) is isomorphic to the symplectic group Sp2⁢n⁢(r) (see [3, Exercise 8.5, p. 116]). Hence Aut⁡(Rn,r) contains a subgroup An,r≅Sp2⁢n⁢(r) that centralizes Z⁢(Rn,r). We can form the natural semidirect product

Take a prime p with p≡1⁢(mod⁡r). It is known (see [22, p. 151]) that the extra-special group Rn,r has a faithful irreducible representation of degree rn over the field 𝔽p of order p. Moreover, this representation extends to a faithful representation of Xn,r (see [2, (3A) and (3B)]). Let Vn,r be a faithful irreducible 𝔽p⁢Xn,r-module of dimension rn corresponding to this representation and form the natural semidirect product

The next assertion lists some basic properties of Gn,r which can be easily deduced from the definition of this group.

For a positive integer n, put

Observe that by definition the set 𝒢n is infinite.

Now we prove Theorem 1. Assume that statement (2) of the Borovik–Khukhro conjecture is true. Fix a positive integer n and consider the set 𝒢n. Theorem 3 implies that the c-dimensions of groups in 𝒢n are bounded by the number given in (2.2). Hence every group Gn,r/(Rn,r⁢Vn,r)≅Sp2⁢n⁢(r) for r∈πn¯ contains an abelian subgroup, and therefore a normal abelian subgroup, of index bounded in terms of n. But there is a unique maximal proper normal subgroup of Sp2⁢n⁢(r), namely its center of order 2, and the index of these subgroups increases with r. This contradiction completes the proof of Theorem 1.

Theorem 1 implies that there exists a positive integer k such that the second part of the Borovik–Khukhro conjecture fails for the finite groups of c-dimension k. On the other hand, the description of the finite groups of c-dimension 2 (see [25, Theorem 9.3.12]) yields the full version of the conjecture for these groups. So it is natural to ask what is the minimal value of the c-dimension for which the Borovik–Khukhro conjecture is false. We already know that 3 is a lower bound for this number and we do not know any better lower bound. Let us obtain an upper bound by bounding M2.

Recall that M2 is the smallest M such that there exist an infinite number of prime numbers r satisfying Ω⁢(r2-1)⩽M.

Since r2-1 is divisible by 24 for r>3, it is clear that Ω⁢(r2-1)⩾6 for sufficiently large r, and therefore M2⩾6. There is a question attributed to P. Neumann [26, p. 316] asking whether or not the number of prime numbers r such that the order of PSL2⁢(r) is a product of six prime numbers is infinite. Since

the affirmative answer to this question would imply that M2=6 and the c-dimensions of groups in 𝒢1 would not exceed 2⋅(22+6)=20.

The affirmative answer to Neumann’s question would follow from the validity of Dickson’s conjecture (the k-tuple conjecture) [9]. This conjecture states that for a finite set of integral linear forms a1⁢n+b1,…,ak⁢n+bk with ai>0, there are infinitely many positive integers n such that all these forms are simultaneously prime unless there are fixed divisors of their product. Equivalently,

If Dickson’s conjecture is valid for the triplet 12⁢n+1, 6⁢n+1 and n, then there are infinitely many primes r of the form r=12⁢n+1 such that n and 6⁢n+1 are also primes and

Both Dickson’s and Neumann’s conjectures provide an exact value of M2, but for our purposes a partial result might be satisfactory. There are well developed sieve methods that allow to obtain partial results for this kind of statement (see [18] for details). For example, the main result of [23] is very close to what is required. In this paper, J. Maynard proved that for any triplet of the forms ai⁢n+bi, i=1,2,3, without fixed divisors of their product, there are infinitely many positive integers n such that

In particular, there are infinite many n such that the product n⋅(6⁢n+1)⋅(12⁢n+1) has at most seven prime divisors, and, therefore, there are infinitely many integers m such that Ω⁢(m⁢(m2-1))⩽11. Unfortunately, one cannot guarantee the primality of m in this result. Nevertheless, in a private letter, J. Maynard expressed confidence that one can show that there are infinitely many primes r such that Ω⁢(r⁢(r2-1))⩽11, or equivalently Ω⁢(r2-1)⩽10, by slightly changing his proof. This would mean that the c-dimensions of groups in 𝒢1 do not exceed 28.

Proposition 2.1, which is also proved in [1] by using sieve methods, has the following refinement [1, Corollary 4.2]: there are infinitely many primes r such that Ω⁢(r2-1)⩽21. This statement implies that M2⩽21, and therefore the following is a consequence of Theorem 3.

It is clear that the number p in the construction of the module V1,r can be chosen so that p would not divide |X1,r|. Then one can apply the description of subgroups in SL2⁢(r)≅Sp2⁢(r)≅A1,r, the character table of this group and the results of [20] to calculate the character of the representation of X1,r corresponding to the module V1,r and use it to calculate the precise value of the c-dimension of G1,r.

3 Proof of Theorem 2

Throughout the rest of the paper G denotes a finite group and k denotes the c-dimension of G. The function λ was defined in Section 1. We start by bounding the value of λ on the layer E⁢(G) of G.

Now we are ready to complete the proof of Theorem 2. Put G¯=G/F3⁢(G). Let F¯* be the generalized Fitting subgroup of G¯. The quotient G¯/F¯* is isomorphic to a subgroup of the group of outer automorphisms Out⁡(F¯*) of F*. The latter is a subgroup of the direct product of Out⁡(F⁢(G¯)) and Out⁡(E⁢(G¯)). By Lemma 1.12, the order of F⁢(G¯) and, therefore, the order of Out⁡(F⁢(G¯)) is bounded in terms of k. Hence it is sufficient to prove that Out⁡(E⁢(G¯)) contains an abelian subgroup of k-bounded index.

If E⁢(G¯) is a product of components Q1,Q2,…,Qs, then Out⁡(E⁢(G)) is a subgroup of a semidirect product of Out⁡(Q1)×Out⁡(Q2)×…×Out⁡(Qs) and some subgroup of Syms. According to Lemma 1.11, the number s is bounded in terms of k, therefore, it is sufficient to prove that the direct product of the outer automorphism groups of the components has an abelian subgroup of k-bounded index. The group Out⁡(Qi) is a subgroup of the outer automorphism group of the corresponding simple group Si=Qi/Z⁢(Qi) (see [15, Corollary 5.1.4]). If Si is an alternating or sporadic group, then the order of Out⁡(Si) is at most 4. If Si is a group of Lie type, then the index of the cyclic subgroup of the field automorphisms in Out⁡(Si) is bounded in terms of the Lie rank of Si (see [15, Theorems 2.5.1 and 2.5.12]). Due to Proposition 3.2, the Lie rank of every Si is bounded in terms of k. Define Ai to be trivial if Si is not a group of Lie type, and the group of the field automorphisms of Si otherwise. The index of the direct product A1×…×As in Out⁡(S1)×…×Out⁡(Ss) is k-bounded by the preceding remarks. This completes the proof of the theorem.

Observe that since the group G¯/Soc⁡(G¯) is a homomorphic image of the quotient group of G/F3⁢(G) by its layer, the following is an immediate consequence of Theorem 2.