Two-sided configuration equivalence and isomorphism

Abstract

The concept of configuration was first introduced by Rosenblatt and Willis to give a characterization for the amenability of groups. Then Rejali and Yousofzadeh introduced the notion of two-sided configuration to study the normal subsets of a group. In [A, Abdollahi, A. Rejali and G. A. Willis, Group properties characterised by configurations, Illinois J. Math. 48(3) (2004) 861–873.], the authors have asked that if two configuration equivalent groups are isomorphic? We show that if [Formula: see text] and [Formula: see text] have same two-sided configuration sets and [Formula: see text] is a normal subgroup of [Formula: see text] with polycyclic or FC quotient, then there is a normal subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text]. Also, we show that if [Formula: see text] and [Formula: see text] are two-sided equivalent groups, and if one of them is polycyclic or FC, then they are isomorphic.