# Homomorph-containment is not transitive

From Groupprops

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

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## Statement

### Statement with symbols

It is possible to have groups such that is a homomorph-containing subgroup of and is a homomorph-containing subgroup of , but is not a homomorph-containing subgroup of .

## Related facts

- Subhomomorph-containment is transitive: Subhomomorph-containment is a stronger subgroup property that is transitive.
- Subhomomorph-containing implies right-transitively homomorph-containing: A homomorph-containing subgroup of a subhomomorph-containing subgroup is homomorph-containing.
- Full invariance is transitive: The property of being a fully invariant subgroup is a weaker subgroup property that is transitive.

## Proof

### An example

Consider the group:

.

Let be the first direct factor and be the center of , which is a cyclic subgroup of order two in . Then:

- is homomorph-containing in : For any homomorphism from to the projection to the second direct factor is trivial since is a perfect group and therefore has no quotient of order two. Thus, any homomorphism from to has image in .
- is homomorph-containing in : In fact, the only element of order two in is the non-identity element of .
- is not homomorph-containing in : There is a a homomorphism from to mapping isomorphically to the second direct factor.