Direct factor implies normal

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., direct factor) must also satisfy the second subgroup property (i.e., normal subgroup)
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Statement

Suppose $G$ is an internal direct product of subgroups $H$ and $K$, so that both $H$ and $K$ are direct factors of $G$. Then, both $H$ and $K$ are normal subgroups of $G$.

Intermediate properties

• Central factor
• Complemented normal subgroup

Proof

The proof is direct from the definition.