Martin J. Evans

On Groups in Which Every Subgroup of Infinite Rank Is Subnormal of Bounded Defect

Abstract A special case of the main result is as follows: Let G be a locally (soluble-by-finite) group of infinite rank in which every subgroup of infinite rank is subnormal of defect at most d. Then G is nilpotent of class bounded in terms of d only. This extends a famous theorem of Roseblade and holds, more generally, for groups G that belong to the class 𝔛 introduced by Černikov. It is also shown that G is a Dedekind group if d = 1.

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 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.

Primitive elements in free groups

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).

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

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