Isomorph-containing iff weakly closed in any ambient group
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 of a group :
- is an isomorph-containing subgroup of : For any subgroup of isomorphic to , .
- For any group containing , is weakly closed in relative to .
- 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
Isomorph-containing implies weakly closed in any ambient group ((1) implies (2))
Given: A group , a subgroup that is isomorph-containing. A group containing .
To prove: is weakly closed in relative to : for any such that , we have .
Proof: Suppose is such that . Then, is isomorphic to , because conjugation by is an automorphism. In particular, because is isomorph-containing in , and we are done.
Weakly closed in ambient group implies isomorph-containing ((2) implies (1))
Given: A group . A subgroup of such that is weakly closed in relative to any group containing .
To prove: If is a subgroup of isomorphic to , then .
Proof: Let be an isomorphism. By fact (1), there exists a group containing and an element such that conjugation by induces on . In particular, we get . Since is weakly closed in relative to , we get , forcing .