# Subisomorph-containing iff strongly closed in any ambient group

From Groupprops

This article gives a proof/explanation of the equivalence of multiple definitions for the term subisomorph-containing subgroup

View a complete list of pages giving proofs of equivalence of definitions

## Statement

The following are equivalent for a subgroup of a group :

- is a subisomorph-containing subgroup of , i.e., if is a subgroup of isomorphic to a subgroup of , then is contained in .
- For any group containing , is a Strongly closed subgroup (?) of with respect to .

## Related facts

- Isomorph-containing iff weakly closed in any ambient group
- Same order iff potentially conjugate
- Isomorphic iff potentially conjugate, isomorphic iff potentially conjugate in finite
- Inner automorphism to automorphism is right tight for normality
- Every injective endomorphism arises as the restriction of an inner automorphism