Isomorph-containing iff weakly closed in any ambient group

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This article gives a proof/explanation of the equivalence of multiple definitions for the term isomorph-containing subgroup
View a complete list of pages giving proofs of equivalence of definitions

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

Facts used

  1. Isomorphic iff potentially conjugate

Proof

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.