Two-sided Configuration of Polycyclic Groups and Isomorphism

The concept of configuration was first introduced to give a condition for the amenability of groups. Then the concept of two-sided configuration was suggested to provide \enquote{normality} to study the group structures more efficiently. We show that if $G$ and $H$ have the same two-sided configuration sets and $N$ is a normal subgroup of $G$ with polycyclic quotient, then there is a normal subgroup $\mathfrak N$ of $H$ such that $G/N\cong H/ \mathfrak N$. Using two-sided configuration equivalence, we prove that being two-sided configuration equivalent to a group $G$ which has a normal sub-series, each of its factor is polycyclic, leads to isomorphism. To show this, we first prove the statement for polycyclic groups. In particular, for the class of nilpotent or FC, the notions of two-sided configuration equivalence and isomorphism are coincided.