# Equivalence of definitions of variety-containing subgroup of finite group

From Groupprops

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 of a finite group :

- The subgroup is a Variety-containing subgroup (?): Let be the subvariety of the variety of groups generated by the group . Then, for any subgroup of in , is contained in .
- The subgroup is a Subhomomorph-containing subgroup (?): contains any subgroup of that occurs as a homomorphic image of a subgroup of .
- The subgroup is a Subisomorph-containing subgroup (?): contains any subgroup of that is isomorphic to a subgroup of .

## 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