  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.
