Subhomomorph-containing not implies variety-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., subhomomorph-containing subgroup) need not satisfy the second subgroup property (i.e., variety-containing subgroup)
View a complete list of subgroup property non-implications | View a complete list of subgroup property implications
Get more facts about subhomomorph-containing subgroup|Get more facts about variety-containing subgroup
EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property subhomomorph-containing subgroup but not variety-containing subgroup|View examples of subgroups satisfying property subhomomorph-containing subgroup and variety-containing subgroup
- Equivalence of definitions of variety-containing subgroup of finite group: For a finite group, the two notions do coincide, and they also coincide with the notion of subisomorph-containing subgroup.
- Equivalence of definitions of variety-containing subgroup of periodic group
- Subisomorph-containing not implies subhomomorph-containing
Let be the direct product of cyclic groups of order for all positive integers . Let be the subgroup of comprising all the elements of finite order.
- is a subhomomorph-containing subgroup of : Any subgroup of is periodic, and the homomorphic image of such a subgroup is thus also periodic. Thus, any homomorphic image of any subgroup of is contained in . So, is a subhomomorph-containing subgroup.
- is not a variety-containing subgroup of : contains each of the direct factors of , because each factor itself has finite order. Thus, each of these is in the subvariety generated by . Hence, so is , their direct product. But is not a subgroup of , since is a proper subgroup of .