Isomorph-containing implies characteristic

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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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Statement

Any isomorph-containing subgroup of a group is a characteristic subgroup.

Definitions used

Isomorph-containing subgroup

Further information: isomorph-containing subgroup

A subgroup H of a group G is termed an isomorph-containing subgroup if, for every subgroup K of G isomorphic to H, K \le H.

Characteristic subgroup

Further information: characteristic subgroup

A subgroup H of a group G is termed a characteristic subgroup if, for every automorphism \sigma of G, \sigma(H) \le H.

Related facts

Stronger facts

Proof

Given: An isomorph-containing subgroup H of a group G.

To prove: For every automorphism \sigma of G, \sigma(H) \le H

Proof: Since \sigma is an automorphism, the restriction of \sigma to H is an isomorphism from H to \sigma(H). Hence, \sigma(H) is isomorphic to H. Since H is isomorph-containing, we get \sigma(H) \le H, completing the proof.