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.

Key Words:

• Finite rank
• Subnormality

Mathematics Subject Classification:

• Primary 20F22
• Secondary 20E07, 20F19

Acknowledgment

The second author would like to thank the Graduate School of the University of Alabama for financial support while part of this work was being done.

Notes

^#Communicated by M. Dixon.