# Subhomomorph-containing not implies variety-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., subhomomorph-containing subgroup) neednotsatisfy the second subgroup property (i.e., variety-containing subgroup)

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## Statement

It is possible to have a group and a subhomomorph-containing subgroup of that is not a variety-containing subgroup of .

## Related facts

- 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

## Proof

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 .