Two-sided configuration equivalence and isomorphism

The concept of configuration was first introduced to give a characterization for the amenability of groups. Then the concept of two-sided configuration was suggested to provide normality to study the group structures more efficiently. It has been interesting that for which groups, two-sided configuration equivalence would imply isomorphism. We introduce a class of groups, containing polycyclic and FC groups, which for them, the notions of two-sided configuration equivalence and isomorphism coincide.