Avinoam Mann

Powerful p-groups. I. Finite groups

In this paper we study a special class of finite p-groups, which we call powerful p-groups. In the second part of this paper, we apply our results to the study of p-adic analytic groups. This application is possible, because a finitely generated pro-p group is p-adic analytic if and only if it is “virtually pro-powerful.” These applications are described in the introduction to the second part, while now we describe the present part in more detail. In the first section we define a powerful p-group, as one whose subgroup of pth powers contains the commutator subgroup. We give several results on these groups, in particular show that many naturally defined subgroups of them are also powerful, and then use this to show that if H is any subgroup of the powerful group G, then the number of generators of H is at most the same number for G. This result and its “converse” can be regarded as one of our main results, the said converse stating that, if all subgroups of the p-group G can be generated by at most Y elements, then G contains a powerful subgroup whose index is bounded by a function of r only. Thus the study of powerful groups is related to the study of “groups of rank r,” i.e., groups with bound r on the number of generators of subgroups, as above. This connection is exploited in Section 2, where we use it, e.g., to establish a conjecture of Jones and Wiegold, that the number of generators of the multiplicator of a group of rank r is bounded by a function of r. Section 3 contains examples and some further results. Some of these relate our results to concepts from the “power-structure” of p-groups, as discussed in a previous paper by the second author [Ma]. There is some difference, both in the definitions and in the results, between the odd primes and the prime 2. Thus, in the first three sections we assume that p is an odd prime, while in the last section we take p = 2. In that section we just repeat the statement of the results of the previous sections, as far as we know them to be true, with the necessary modifications. We give proofs only if they differ from the ones for odd primes. The result corresponding to 1.8, e.g., is numbered 4.1.8, etc. 484 OC21-8693/87

Finitely generated groups of polynomial subgroup growth

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Powerful p-groups. II. p-adic analytic groups

We determine the structure of finitely generated residually finite groups in which the number of subgroups of each finite indexn is bounded by a fixed power ofn.

On groups of polynomial subgroup growth

We apply our results from the first part [LM] to p-adic analytic pro-p groups, i.e., pro-p groups which are Lie groups over the field of p-adic numbers. For a systematic study of these groups see [La, B, Sl]. Lazard [La] characterized the pro-p groups which are p-adic analytic as (in the terminology of [LM]) the finitely generated virtually powerful pro-p groups. Our detailed study of finite powerful groups enables us to get a new characterization:

Injectors and normal subgroups of finite groups

SummaryLet Γ be a finitely generated group andan(Γ)=the number of its subgroups of indexn. We prove that, assuming Γ is residually nilpotent (e.g., Γ linear), thenan(Γ) grows polynomially if and only if Γ is solvable of finite rank. This answers a question of Segal. The proof uses a new characterization ofp-adic analytic groups, the theory of algebraic groups and the Prime Number Theorem. The method can be applied also to groups of polynomial word growth.

GENERALIZED FROBENIUS GROUPS. II

A class of subgroups,N-injectors, previously defined only for soluble groups, is here defined for some non-soluble groups. It is proved that injectors have some properties which resemble those of Sylow subgroups.

Simple groups, maximal subgroups, and probabilistic aspects of profinite groups

A pair (G. K) in whichG is a finite group andK◃G, 1<K<G, is said to satisfy (F2) if |CG(x)|=|CG/K(xK)| for allx∈G/K. First we survey all the examples known to us of such pairs in whichG is neither ap-group nor a Frobenius group with Frobenius kernelK. Then we show that under certain restrictions there are, essentially, all the possible examples.

Residually finite groups of finite rank

We show that a finite simple group has at mostn1.875+o(1) maximal subgroups of indexn. This enables us to characterise profinite groups which are generated with positive probability by boundedly many random elements. It turns out that these groups are exactly those having polynomial maximal subgroup growth. Related results are also established.

Conjugacy classes in finite groups

The recent constructions, by Rips and Olshanskii, of infinite groups with all proper subgroups of prime order, and similar ‘monsters’, show that even under the imposition of apparently very strong finiteness conditions, the structure of infinite groups can be rather weird. Thus it seems reasonable to impose the type of condition that enables us to apply the theory of finite groups. Two such conditions are local finiteness and residual finiteness, and here we are interested in the latter. Specifically, we consider residually finite groups of finite rank, where a group is said to have rank r , if all finitely generated subgroups of it can be generated by r elements. Recall that a group is said to be virtually of some property, if it has a subgroup of finite index with this property. We prove the following result: Theorem 1. A residually finite group of finite rank is virtually locally soluble.

Some questions about p -groups

In the first part of this note, we give new proofs of known results regarding the class number of finite groups, adding a few related results. In the second part, we improve a result of Ito concerning a special class ofp-groups.