Corollary of Krull-Remak-Schmidt theorem for cancellation of powers

Statement
Suppose $$G$$ and $$H$$ are groups such that $$G$$ satisfies the following two conditions:


 * the ascending chain condition on normal subgroups
 * the descending chain condition on normal subgroups

Suppose $$m$$ is a positive integer such that the direct power $$G^m$$ (the external direct product of $$m$$ copies of $$G$$) is isomorphic to the direct power $$H^m$$.

Then, $$G$$ and $$H$$ are isomorphic.

In particular, this result holds when $$G$$ is a finite group.

Facts used

 * 1) uses::Krull-Remak-Schmidt theorem