Isomorph-containing iff weakly closed in any ambient group

The definitions that we have to prove as equivalent
The following are equivalent for a subgroup $$H$$ of a group $$G$$:


 * 1) $$H$$ is an isomorph-containing subgroup of $$G$$: For any subgroup $$K$$ of $$G$$ isomorphic to $$H$$, $$K \le H$$.
 * 2) For any group $$L$$ containing $$G$$, $$H$$ is weakly closed in $$G$$ relative to $$L$$.

Related facts

 * Same order iff potentially conjugate
 * Isomorphic iff potentially conjugate
 * Left transiter of normal is characteristic
 * Inner automorphism to automorphism is right tight for normality
 * Every injective endomorphism arises as the restriction of an inner automorphism
 * Subisomorph-containing iff strongly closed in any ambient group

Facts used

 * 1) uses::Isomorphic iff potentially conjugate

Isomorph-containing implies weakly closed in any ambient group ((1) implies (2))
Given: A group $$G$$, a subgroup $$H$$ that is isomorph-containing. A group $$L$$ containing $$G$$.

To prove: $$H$$ is weakly closed in $$G$$ relative to $$L$$: for any $$g \in L$$ such that $$gHg^{-1} \le G$$, we have $$gHg^{-1} \le H$$.

Proof: Suppose $$g \in L$$ is such that $$gHg^{-1} \le G$$. Then, $$gHg^{-1}$$ is isomorphic to $$H$$, because conjugation by $$g$$ is an automorphism. In particular, $$gHg^{-1} \le H$$ because $$H$$ is isomorph-containing in $$G$$, and we are done.

Weakly closed in ambient group implies isomorph-containing ((2) implies (1))
Given: A group $$G$$. A subgroup $$H$$ of $$G$$ such that $$H$$ is weakly closed in $$G$$ relative to any group $$L$$ containing $$G$$.

To prove: If $$K$$ is a subgroup of $$G$$ isomorphic to $$H$$, then $$K \le H$$.

Proof: Let $$\sigma:H \to K$$ be an isomorphism. By fact (1), there exists a group $$L$$ containing $$G$$ and an element $$g \in L$$ such that conjugation by $$g$$ induces $$\sigma$$ on $$H$$. In particular, we get $$gHg^{-1} = K \le G$$. Since $$H$$ is weakly closed in $$G$$ relative to $$L$$, we get $$gHg^{-1} \le H$$, forcing $$K \le H$$.