Isomorph-containing subgroup

Definition
A subgroup $$H$$ of a group $$G$$ is termed an isomorph-containing subgroup if it satisfies the following equivalent conditions:


 * 1) Whenever $$K \le G$$ is a subgroup of $$G$$ isomorphic to $$H$$, $$K \le H$$.
 * 2) If $$G$$ is a subgroup of $$L$$, $$H$$ is weakly closed in $$G$$ with respect to $$L$$.

Stronger properties

 * Weaker than::Isomorph-free subgroup: For a finite subgroup, and more generally, for a co-Hopfian subgroup, the two properties are equivalent.
 * Weaker than::Homomorph-containing subgroup: Also related:
 * Weaker than::Subhomomorph-containing subgroup
 * Weaker than::Subisomorph-containing subgroup
 * Weaker than::Variety-containing subgroup
 * Weaker than::Fully invariant direct factor:

Weaker properties

 * Stronger than::Characteristic subgroup: Also related:
 * Stronger than::Intermediately injective endomorphism-invariant subgroup
 * Stronger than::Injective endomorphism-invariant subgroup
 * Stronger than::Intermediately characteristic subgroup
 * Stronger than::Normal subgroup
 * Stronger than::Normal-isomorph-containing subgroup