Direct factor satisfies intermediate subgroup condition

This article gives the statement, and possibly proof, of a subgroup property (i.e., direct factor) satisfying a subgroup metaproperty (i.e., intermediate subgroup condition)
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Statement

Suppose $H\le K\le G$ are groups such that $H$ is a direct factor of $G$. Then, $H$ is also a direct factor of $K$.

Related facts

• Permutably complemented satisfies intermediate subgroup condition
• Complemented normal satisfies intermediate subgroup condition
• Retract satisfies intermediate subgroup condition