Homomorph-containment satisfies intermediate subgroup condition

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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

Proof

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

To prove: \varphi(H) is contained in H.

Proof: Since K \le G, we can compose \varphi with the inclusion of K in G to get a homomorphism \varphi':H \to G. Since H is homomorph-containing in G, \varphi'(H) \le H, so \varphi(H) \le H.