B. Hartley

Finite and Locally Finite Groups

Preface. Introduction. Simple locally finite groups B. Hartley. Algebraic groups G.M. Seitz. Subgroups of simple algebraic groups and related finite and locally finite groups of Lie type M.W. Liebeck. Finite simple groups and permutation groups J. Saxl. Finitary linear groups: a survey R.E. Phillips. Locally finite simple groups of finitary linear transformations J.I. Hall. Non-finitary locally finite simple groups U. Meierfrankenfeld. Inert subgroups in simple locally finite groups V.V. Belyaev. Group rings of simple locally finite groups A.E. Zalesskii. Simple locally finite groups of finite Morley rank and odd type A.V. Borovik. Existentially closed groups in specific classes F. Leinen. Groups acting on polynomial algebras R.M. Bryant. Characters and sets of primes for solvable groups I.M. Isaacs. Character theory and length problems A. Turull. Finite p-groups A. Shalev. Index.

Subgroups of finite index in profinite groups

Let ~ be a class of finite groups. By this we understand that ~ is a class in the usual sense, which contains all groups of order 1, and contains, with every group G ~ , all isomorphic copies of G. By a pro-~ group, we mean a topological group isomorphic to an inverse limit of groups in ~, viewed as a topological group in the usual way. If ~ is closed under taking homomorphic images, this is equivalent to saying that G is a compact totally disconnected Hausdorff topological group such that G/N~g for every open normal subgroup N of G. We write g* for the class of all pro-~ groups. It seems to be unknown whether every subgroup of finite index in a finitely generated profinite group is open. Here we say that a profinite group is finitely generated, if it has a dense subgroup which is finitely generated in the algebraic sense. The answer is known to be affirmative if ~ is the class 919l of finite abelian-by-nilpotent groups (Anderson [1]) or the class of finite supersoluble groups (Oltikar and Ribes [6]). I am indebted to L. Ribes for bringing these results to my attention, and for several stimulating discussions. We generalize these results as follows. For an integer l> 1, let 91l denote the class of all finite groups G which have a series

Periodic locally soluble groups with the minimal condition on centralizers

In a recent paper [l] the first author studied locally finite groups satisfying the minimal condition on centralizers of subsets. One of the main results was that, for every prime p, the Sylow P-subgroups of such a group are conjugate. In the present paper it is shown that if G is a periodic soluble group satisfying the minimal condition on centralizers, then, for every set r of primes, the Sylo~ n-subgroups (by which we simply mean maximal r-subgroups) of G are conjugate. It follows that G belongs to the class 21 introduced in [4], and hence the structure of G is constrained by the results of [5] and [8]. In this vein we shall show that G is nilpotent-by-Abelian-by-finite. Our other main result is that a periodic locally soluble group satisfying the minimal condition on centralizers is soluble. These results are generalizations of known results for linear groups (see [13]), and, more generally, for CZ-groups (see [2, 3, 9, IO]). However we conclude with an example to show the limitations of the comparison with linear groups.

Centralizers of finite nilpotent subgroups in locally finite groups

We investigate the structure of locally finite groups with a finite subgroup whose centralizer is close to a linear group.

Simple Locally Finite Groups

Beginning from basic principles, we outline the current state of affairs in the theory of locally finite simple groups. Particular emphasis is placed on constructions, Kegel sequences, and centralizers.

On the quotient of a free group by the commutator of two normal subgroups

Abstract Let F be a noncyclic free group and let K and L be normal subgroups of F . In trying to describe the group F [K, L] in terms of F K and F L , it is important to understand the group A=K∩ F [K, L] . The main theorem gives an exact sequence into which the group A fits. The sequence reduces in special cases to Hopfs Formula for H 2 ( F K, Z ) and to the Magnus embedding or relation sequence embedding the relation module in a free module. The sequence is 0→H 2 (S)⊕H 2 (T)→A→P→Δ(C)⊗ zc ZB→0 . where B= F KL , C= F K ∩L, Δ(C) denotes the augmentation ideal of C , P is a free ZB -module on a basis in one to one correspondence with a basis of F , and the sequence is of ZB -modules. As one application of this, we show that if each of K and L is the normal closure of a single element, then A is free abelian. A second application gives that if l >2 and F n , F ( n ) are the n th terms of the lower central and derived series respectively of F , then F [F … F (f) ] contains an infinite elementary abelian 2-group.

On modular relation modules and residual nilpotence

Let F be a non-cyclic free group, let R d F, and let G = FIR. In [3], Gruenberg showed that if G is finite, then FIR’ is residually nilpotent if and only if G is a finite p-group. In that paper, the connection between this question and the intersection of the powers of the augmentation ideal 0 of the integral group ring ZG, was brought out. It was natural then for the same question to be taken up for infinite G, and this was done by several authors, culminating in the paper of Lichtman [14]. There, completely general conditions on G were given, equivalent to the residual nilpotence of F/R’. In [7], this work was extended to give necessary and sufficient conditions for F/S to be residually nilpotent, where S is a fully invariant subgroup of R such that R/S is a non-trivial torsion-free nilpotent group. For more details on these matters, see [g]. It seems natural to investigate the same question when R/S is not torsion-free, and in particular when it is a relatively free nilpotent p-group, and this is one of the themes of this paper. Much remains to be said here, but we shall prove the following result. If H is a group and p a prime, we write D,,(H, p”‘) for the nth dimension subgroup of H over the ring of integers modulo p’“. This is clearly a fully invariant subgroup of H.