Homomorph-containment satisfies intermediate subgroup condition

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

Verbal statement

A homomorph-containing subgroup of the whole group is also a homomorph-containing subgroup of any intermediate subgroup.

Statement with symbols

Suppose $H\le K\le G$ are groups, and $H$ is a homomorph-containing subgroup of $G$. Then, $H$ is also a homomorph-containing subgroup of $K$.

Related facts

• Full invariance does not satisfy intermediate subgroup condition
• Homomorph-containing implies intermediately fully invariant
• Homomorph-containment does not satisfy image condition

Proof

Given: Groups $H\le K\le G$ such that $H$ is homomorph-containing in $G$. A homomorphism $\phi :H\to K$.

To prove: $\phi (H)$ is contained in $H$.

Proof: Since $K\le G$, we can compose $\phi$ with the inclusion of $K$ in $G$ to get a homomorphism ${\phi }^{\prime }:H\to G$. Since $H$ is homomorph-containing in $G$, ${\phi }^{\prime }(H)\le H$, so $\phi (H)\le H$.