# Homomorph-containment satisfies intermediate subgroup condition

From Groupprops

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 are groups, and is a homomorph-containing subgroup of . Then, is also a homomorph-containing subgroup of .

## 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 such that is homomorph-containing in . A homomorphism .

**To prove**: is contained in .

**Proof**: Since , we can compose with the inclusion of in to get a homomorphism . Since is homomorph-containing in , , so .