Groups with normality conditions for subgroups of infinite rank

Abstract

A well-known theorem of B. H. Neumann states that a group has finite conjugacy classes of subgroups if and only if it is central-by-finite. It is proved here that if G is a generalized radical group of infinite rank in which the conjugacy classes of subgroups of infinite rank are finite, then every subgroup of G has finitely many conjugates, and so G/Z(G) is finite. Corresponding results are proved for groups in which every subgroup of infinite rank has finite index in its normal closure, and for those in which every subgroup of infinite rank is finite over its core.

Groups of infinite rank with finite conjugacy classes of subnormal subgroups

“…The study of the influence on a soluble group of the behavior of its subgroups of infinite rank has been developed in a series of recent papers (see for instance [3][4][5][6][7][8][9][10]); it suggests that in a group of infinite rank the behavior of subgroups of finite rank with respect to an embedding property can be neglected. In particular, it was proved in [7] that if G is a periodic soluble group of infinite rank in which every subnormal subgroup of infinite rank is normal, then G is a T -group.…”

Groups With Normality Conditions for Uncountable Subgroups

“…In a famous paper of 1955, Neumann [11] proved that all subgroups of a group G are almost normal if and only if the centre Z(G) has finite index, and that groups admitting only nearly normal subgroups are precisely those with a finite commutator subgroup. The imposition of almost normality or near normality only on the members of a relevant system of subgroups also gives rise to strong restrictions on the group structure (see, for instance, [2,8,9]).…”

On groups whose subgroups of infinite rank are Sylow permutable

“…The results of the present paper analyse the impact of the embedding of the subgroups of infinite section rank on the structure of a periodic group and can be viewed as belonging to a larger family of results stating that in a group of infinite special rank the behaviour of subgroups of finite special rank with respect to a given subgroup embedding property can be ignored (see [4][5][6][7][8][10][11][12][13]). …”