Equivalence of definitions of variety-containing subgroup of finite group

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

• Equivalence of definitions of variety-containing subgroup of periodic group: The proof can be generalized to any periodic group, i.e., a group where every element has finite order.
• omega subgroups not are variety-containing
• Variety-containing implies omega subgroup in group of prime power order

Opposite facts

• Subisomorph-containing not implies subhomomorph-containing
• Subhomomorph-containing not implies variety-containing