Equivalence of definitions of variety-containing subgroup of finite group

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This article gives a proof/explanation of the equivalence of multiple definitions for the term variety-containing subgroup of finite group
View a complete list of pages giving proofs of equivalence of definitions

Statement

The following are equivalent for a subgroup H of a finite group G:

  1. The subgroup is a Variety-containing subgroup (?): Let \mathcal{V} be the subvariety of the variety of groups generated by the group H. Then, for any subgroup K of G in \mathcal{V}, K is contained in H.
  2. The subgroup is a Subhomomorph-containing subgroup (?): H contains any subgroup of G that occurs as a homomorphic image of a subgroup of H.
  3. The subgroup is a Subisomorph-containing subgroup (?): H contains any subgroup of G that is isomorphic to a subgroup of H.

Related facts

Opposite facts