ACIC is characteristic subgroup-closed

This article gives the statement, and possibly proof, of a group property (i.e., ACIC-group) satisfying a group metaproperty (i.e., characteristic subgroup-closed group property)
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Statement

Suppose $G$ is an ACIC-group, i.e., every automorph-conjugate subgroup of $G$ is characteristic in $G$ (note that characteristic implies automorph-conjugate, so in such a group, the notion of being a characteristic subgroup precisely coincides with the notion of being an automorph-conjugate subgroup).

Suppose $H$ is a characteristic subgroup (or equivalently, an automorph-conjugate subgroup) of $G$. Then, $H$ is also an ACIC-group.

Generalizations

• Transitive implies intermediate subgroup condition implies closed under former

Definitions used

Term	Definitions
automorph-conjugate subgroup	A subgroup $H$ of a group $G$ is termed automorph-conjugate if any subgroup of $G$ automorphic to $H$ is conjugate to $H$.
ACIC-group	A group $G$ is termed ACIC if every automorph-conjugate subgroup of $G$ is characteristic in $G$; equivalently, every automorph-conjugate subgroup of $G$ is normal in $G$.

Facts used

1. Automorph-conjugacy is transitive
2. Normality satisfies intermediate subgroup condition

Proof

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Given: An ACIC-group $G$, a characteristic subgroup (or equivalently, an automorph-conjugate subgroup) $H$ of $G$, and an automorph-conjugate subgroup $K$ of $H$.

To prove: $K$ is normal in $H$, or equivalently, $K$ is characteristic in $H$.

Proof:

Step no.	Assertion/construction	Facts used	Given data used	Previous steps used	Explanation
1	$K$ is automorph-conjugate in $G$.	Fact (1)	$H$ is automorph-conjugate in $G$, $K$ is automorph-conjugate in $H$.		Fact-given direct.
2	$K$ is normal in $G$.		$G$ is ACIC.	Step (1)	Step-given direct.
3	$K$ is normal in $H$.	Fact (2)	$K\leq H\leq G$.	Step (2)	Step-given-fact direct.