  Giving a condition for the the amenability of groups, Rosenblatt and Willis,
first introduced the concept of configuration. From the beginning of the
theory, the question whether the concept of configuration equivalence coincides
with the concept of group isomorphism was posed. We negatively answer this open
question by introducing two non-isomorphic, solvable and hence amenable groups
which are configuration equivalent. Also, we will study some types of subgroups
in configuration equivalent groups. In particular, we will prove this
conjecture, due to Rosenblatt and Willis, that configuration equivalent groups,
both include the free non-Abelian group of same rank or not. Finally, we prove
that two-sided equivalent groups have same class numbers.
