Characteristic implies automorph-conjugate

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 must also satisfy the second subgroup property
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Statement

Verbal statement

Any characteristic subgroup of a group is an automorph-conjugate subgroup.

Symbolic statement

Let $H$ be a characteristic subgroup of $G$ . Then $H$ is an automorph-conjugate subgroup of $G$ .

Property-theoretic statement

The subgroup property of being characteristic is stronger than the subgroup property of being automorph-connjugate.

Definitions used

Characteristic subgroup

A subgroup of a group is termed a characteristic subgroup if any automorphism of the group maps the subgroup to itself.

That is, $H \leq G$ is characteristic if for any automorphism $σ$ of $G$ , $σ (H) = H$ .

Automorph-conjugate subgroup

A subgroup of a group is termed an automorph-conjugate subgroup if any automorphism of the group maps the subgroup to a conjugate subgroup.

That is $H \leq G$ is automorph-conjugate if for any automorphism $σ$ of $G$ there exists $g \in G < m a t h > s u c h t h a t < m a t h > σ (H) = g H g^{- 1}$ .

Proof

If $H$ is a characteristic subgroup of $G$ , then for any automorphism $σ$ of $G$ , $σ (H) = H$ . Now, $H$ is a conjugate subgroup to itself, and hence $H$ is automorph-conjugate.