Subhomomorph-containing not implies variety-containing

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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)
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Statement

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

Related facts

Proof

Let G be the direct product of cyclic groups of order n for all positive integers n. Let H be the subgroup of G comprising all the elements of finite order.

  • H is a subhomomorph-containing subgroup of G: Any subgroup of H is periodic, and the homomorphic image of such a subgroup is thus also periodic. Thus, any homomorphic image of any subgroup of H is contained in H. So, H is a subhomomorph-containing subgroup.
  • H is not a variety-containing subgroup of G: H contains each of the direct factors of G, because each factor itself has finite order. Thus, each of these is in the subvariety generated by H. Hence, so is G, their direct product. But G is not a subgroup of H, since H is a proper subgroup of G.