  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
$\frac{G_1}{H_1}\cong\frac{G_2}{H_2}.$ Also configuration of FC-groups and
isomorphism is studied.
