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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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Contents 1 Statement 2 Definitions used 2.1 Isomorph-containing subgroup 2.2 Characteristic subgroup 3 Related facts 3.1 Stronger facts 4 Proof

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, .

Characteristic subgroup

Further information: characteristic subgroup

A subgroup H of a group G is termed a characteristic subgroup if, for every automorphism σ of G, .

Related facts

Stronger facts

• Isomorph-containing implies intermediately characteristic
• Isomorph-containing implies injective endomorphism-invariant
• Isomorph-containing implies intermediately injective endomorphism-invariant

Proof

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

To prove: For every automorphism σ of G,

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