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

Statement

If $G$, $H$, and $K$ are groups of finite composition length such that the external direct products $G \times H$ and $G \times K$ are isomorphic, i.e., $G \times H \cong G \times K$, then $H \cong K$.

Related facts

• Krull-Remak-Schmidt theorem
• Corollary of Krull-Remak-Schmidt theorem for cancellation of powers
• Direct product is cancellative for finite groups

Facts used

1. Krull-Remak-Schmidt theorem