Martyn R. Dixon

Locally (soluble-by-finite) groups with all proper non-nilpotent subgroups of finite rank

Abstract A group G is said to have finite rank r if every finitely generated subgroup of G is at most r-generator. If c is a positive integer we let R c denote the class of nilpotent groups of class at most c, and R c ∗ the class of groups in which every proper non- R c subgroup has finite rank. Our main theorem shows that if G is a locally (soluble-by-finite) group in the class R c ∗ then either G is nilpotent of class at most c or G has finite rank. An analogous result holds for locally soluble ( U 2)∗-groups, where U 2 denotes the class of metabelian groups. We give an example to show that locally finite ( U 2)∗-groups need neither have finite rank nor be metabelian.

Groups with all subgroups permutable or of finite rank

In this paper we investigate the structure of X-groups in which every subgroup is permutable or of finite rank. We show that every subgroup of such a group is permutable.

Groups with all proper subgroups (finite rank)-by-nilpotent

Abstract. Let G be a group in which every proper subgroup is an extension of a group of finite (Prüfer) rank by a nilpotent group of class at most c. We show that if G is locally soluble-by-finite, then G is an extension of a group of finite rank by a nilpotent group of class at most c. This result is extended to cover groups G that belong to a certain extensive class

Embedding groups in locally (soluble-by-finite) simple groups

\frak X

Groups with all proper subgroups nilpotent-by-finite rank

of locally graded groups.

On totally inert simple groups

Abstract The authors investigate the structure of locally (soluble-by-finite) simple groups Ĝ and show that such groups are locally residually finite. Sufficient conditions are given for a group G to be embeddable in such a group Ĝ and an example of a locally (abelian-by-finite) simple group is constructed.

Some countably recognizable classes of groups

Abstract. In this paper the authors consider the class of groups in which every proper subgroup is nilpotent-by-finite rank. There exist infinite simple groups with this property. Among the results proved is the theorem that a locally soluble-by-finite such group that is not a perfect p-group is itself nilpotent-by-finite rank, provided the group has no infinite simple images.

GROUPS WITH ALL PROPER SUBGROUPS (FINITE RANK)-BY-NILPOTENT. II

A subgroup H of a group G is inert if |H: H ∩ Hg| is finite for all g ∈ G and a group G is totally inert if every subgroup H of G is inert. We investigate the structure of minimal normal subgroups of totally inert groups and show that infinite locally graded simple groups cannot be totally inert.

LOCALLY (SOLUBLE-BY-FINITE) GROUPS WITH VARIOUS RESTRICTIONS ON SUBGROUPS OF INFINITE RANK

Abstract We show that the class of groups that are soluble-by-(finite rank) is countably recognizable. Also, if every countable subgroup of the group G is (derived length d)-by-(rank r) then G is (derived length d*)-by-(rank r*) for some bounded d*, r*. Similar results hold for groups that are nilpotent-by-(finite rank).

Groups with Various Minimal Conditions on Subgroups

The authors obtain results concerning the structure of locally soluble-by-finite groups with all proper subgroups (finite rank)-by- nilpotent.