# Subhomomorph-containment is transitive

From Groupprops

This article gives the statement, and possibly proof, of a subgroup property (i.e., subhomomorph-containing subgroup) satisfying a subgroup metaproperty (i.e., transitive subgroup property)

View all subgroup metaproperty satisfactions | View all subgroup metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for subgroup properties

Get more facts about subhomomorph-containing subgroup |Get facts that use property satisfaction of subhomomorph-containing subgroup | Get facts that use property satisfaction of subhomomorph-containing subgroup|Get more facts about transitive subgroup property

## Statement

Suppose . Suppose is a subhomomorph-containing subgroup of and is a subhomomorph-containing group of . Then, is a subhomomorph-containing subgroup of .

## Related facts

- Homomorph-containment is not transitive
- Subhomomorph-containing implies right-transitively homomorph-containing
- Full invariance is transitive

## Proof

**Given**: Groups . A homomorphism for a subgroup of .

**To prove**: is a subgroup of .

**Proof**: Since is a subgroup of , is a subgroup of . Thus, is a homomorphism from a subgroup of . Since is subhomomorph-containing in , is contained in . Thus, can be viewed as a map from to .

Thus, is a map from a subgroup of . Since is subhomomorph-containing in , , and we are done.