# Left-transitively homomorph-containing not implies subhomomorph-containing

From Groupprops

This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., left-transitively homomorph-containing subgroup) neednotsatisfy the second subgroup property (i.e., subhomomorph-containing subgroup)

View a complete list of subgroup property non-implications | View a complete list of subgroup property implications

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EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property left-transitively homomorph-containing subgroup but not subhomomorph-containing subgroup|View examples of subgroups satisfying property left-transitively homomorph-containing subgroup and subhomomorph-containing subgroup

## Related facts

## Proof

Let be the direct product:

where is the alternating group of degree five and is the cyclic group of order two. Let be the first direct factor.

Then:

- For any group containing as a homomorph-containing subgroup, also contains as a homomorph-containing subgroup: For any homomorphism , we can extend it to a homomorphism , since is a direct factor of . Since is homomorph-containing in , , so . Now, since is homomorph-containing in , is contained in .
- is not a subhomomorph-containing subgroup of : has cyclic subgroups of order two, that are isomorphic to cyclic subgroups of order two in and outside .