Isomorph-containing implies characteristic
This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., isomorph-containing subgroup) must also satisfy the second subgroup property (i.e., characteristic subgroup)
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Further information: isomorph-containing subgroup
A subgroup of a group is termed an isomorph-containing subgroup if, for every subgroup of isomorphic to , .
Further information: characteristic subgroup
A subgroup of a group is termed a characteristic subgroup if, for every automorphism of , .
- Isomorph-containing implies intermediately characteristic
- Isomorph-containing implies injective endomorphism-invariant
- Isomorph-containing implies intermediately injective endomorphism-invariant
Given: An isomorph-containing subgroup of a group .
To prove: For every automorphism of ,
Proof: Since is an automorphism, the restriction of to is an isomorphism from to . Hence, is isomorphic to . Since is isomorph-containing, we get , completing the proof.