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