# Right-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., right-transitively homomorph-containing subgroup) neednotsatisfy the second subgroup property (i.e., subhomomorph-containing subgroup)

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## 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:

- Every homomorph-containing subgroup of is a homomorph-containing subgroup of : First, note that itself is homomorph-containing in , because, since is simple, no homomorphic image of can project nontrivially onto the second direct factor. Since is simple, the only homomorph-containing subgroups of are and the trivial subgroup. Both of these are homomorph-containing in <amth>G</math>.
- is not subhomomorph-containing in : has cyclic subgroups of order two, that are isomorphic to cyclic subgroups of order two outside , so is not subhomomorph-containing in .