Left-transitively homomorph-containing not implies subhomomorph-containing
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) need not satisfy the second subgroup property (i.e., subhomomorph-containing subgroup)
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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
Let be the direct product:
- 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 .