Corollary of Krull-Remak-Schmidt theorem for cancellation of direct factors

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Statement

Suppose G,H,K are groups such that both G and H satisfy the ascending chain condition on normal subgroups and descending chain condition on normal subgroups. Then, the external direct product G \times H is isomorphic to the external direct product G \times K, we must also have that H and K are isomorphic groups.

Related facts

Weaker facts

Facts used

  1. Krull-Remak-Schmidt theorem