The concept of configuration was first introduced by Rosenblatt and Willis to give a condition for amenability of groups. We show that if G 1 and G 2 have the same configuration sets and H 1 is a normal subgroup of G 1 with abelian quotient, then there is a normal subgroup H 2 of G 2 such that. Also configuration of FC-groups and isomorphism is studied.

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In this paper, we define a new type of configurations as two-sided configurations, and investigate which group properties can be characterized by them. It is proved that for polycyclic torsion free groups, having the same finite quotient sets does not imply the (two-sided) configuration equivalence. We show that isomorphisms and configuration equivalences coincide for some free products of groups and a class of nilpotent groups.

In this paper, we introduce the concept of configuration graph and show how one can use this notion to simplify the theorem proved by Rejali and Yousofzadeh [Configuration of groups and paradoxical decompositions, Bull. Belg. Math. Soc. Simon Stevin18 (2011) 157–172].