R. M. Bryant

The verbal topology of a group

We define for any group G a topology on G called the verbal topology of G and study groups which satisfy the minimal condition on the closed sets in the verbal topology, called min-cbsed. The main results of the paper establish with easy proofs that certain classes of groups satisfy min-closed. Since it is shown that groups with min-closed are C&groups (see [2] or [5]), these results give some new examples of CZ-groups: for example, finitely generated Abelian-bynilpotent groups. Also, since centralizers are closed in the verbal topology, another consequence is an easy proof of the fact that finitely generated Abelianby-nilpotent groups satisfy the minimal condition on centralizers (or, equivalently, the maximal condition on centralizers). This was proved in a stronger form, but with a rather difficult proof, by Lennox and Roseblade [3], answering a question of P. Hall. The investigation which led to the results mentioned above began with a search for discriminating groups (see Neumann [4]). In this connection we prove that every group with min-closed has a discriminating subgroup of finite index.

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.

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.

The Decomposition of Lie Powers

Let

On the module structure of free Lie algebras

G

Automorphism groups of relatively free groups

be a group,

Automorphisms of relatively free nilpotent groups of infinite rank

F

Free lie algebras and adams operations

a field of prime characteristic

Dense subgroups of the automorphism groups of free algebras

p

Free Lie algebras and formal power series

and