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

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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