# 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.

## 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 $G$.

## 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) $K \le H \le G$. Step (2) Step-given-fact direct.