Groups with Polycyclic-by-Finite Conjugate Classes of Subgroups

Abstract

Neumann characterized the groups in which every subgroup has finitely many conjugates only as central-by-finite groups. If 𝔛 is a class of groups, a group G is said to have 𝔛-conjugate classes of subgroups if G/Core _G (N _G (H)) ∈ 𝔛 for every subgroup H of G. In this paper, we generalize Neumann's result by showing that a group has polycyclic-by-finite classes of conjugate subgroup if and only if it is central-by-(polycyclic-by-finite).

Key Words:

• Conjugacy class
• Polycyclic-by-finite group

AMS Classification:

• 20C07
• 20D10
• 20F24

Acknowledgment

The authors wish to thank the referee for his/her useful suggestions, which have allowed to improve this paper. This research was supported by Proyecto BFM2001-2452 of CICYT (Spain), Proyecto 100/2001 of Gobierno de Aragón (Spain) and the University of Athens-R.C., grant No 3403.

Notes

^#Communicated by M. Dixon.