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From Groupprops

This article gives the statement, and possibly proof, of a group property satisfying a group metaproperty
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Contents 1 Statement 1.1 Property-theoretic statement 1.2 Verbal statement 2 Generalizations 3 Definitions used 3.1 Automorph-conjugate subgroup 3.2 ACIC-group 4 Proof

Statement

Property-theoretic statement

The group property of being an ACIC-group satisfies the group metaproperty of being characteristic subgroup-closed.

Verbal statement

Suppose a group is ACIC: any automorph-conjugate subgroup of it is characteristic. Then, any characteristic subgroup of the group, is also ACIC as an abstract group.

Equivalently, any automorph-conjugate subgroup is ACIC. (The two statements are equivalent because automorph-conjugate subgroups are the same as characteristic subgroups).

Generalizations

• Transitive implies intermediate subgroup condition implies closed under former

Definitions used

Automorph-conjugate subgroup

Further information: automorph-conjugate subgroup

A subgroup H in a group G is termed automorph-conjugate if for every automorphism σ of G, H and σ(H) are conjugate subgroups.

ACIC-group

Further information: ACIC-group

A group is ACIC if it satisfies the following equivalent conditions:

• Every automorph-conjugate subgroup is characteristic
• Every automorph-conjugate subgroup is normal

The two definitions ar eequivalent because being both normal and automorph-conjugate is equivalent to being characteristic.

Proof

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

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

Proof:

• Since automorph-conjugacy is transitive, we see that K is automorph-conjugate, not just in H, but also in G.
• Since G is ACIC, this tells us that K is characteristic in G. In particular, K is normal in G.
• Since normality satisfies intermediate subgroup condition, K is normal in H as well.