ACIC is characteristic subgroup-closed

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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

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