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. Krull-Remak-Schmidt theorem