ACIC implies nilpotent (finite groups)

This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property must also satisfy the second group property
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Statement

Property-theoretic statement

For finite groups, the property of being ACIC is stronger than the property of being nilpotent.

Verbal statement

Any finite ACIC-group is nilpotent.

Definitions used

Automorph-conjugate subgroup

Further information: automorph-conjugate subgroup

A subgroup $H$ of a group $G$ is termed an automorph-conjugate subgroup if for every automorphism $\sigma$ of $G$ , $H$ and $\sigma (H)$ are conjugate subgroups.

ACIC-group

Further information: ACIC-group

A group is termed ACIC if every automorph-conjugate subgroup is characteristic, or equivalently, any automorph-conjugate subgroup is normal.

Finite nilpotent group

Further information: Finite nilpotent group

A finite group is termed nilpotent if all Sylow subgroups are normal (this is just one of the formulations of nilpotence for finite groups). The definition breaks down for infinite groups).

Facts used

In a finite group, any Sylow subgroup is automorph-conjugate. Further information: Sylow implies automorph-conjugate
The definition of finite nilpotent group given above: a finite nilpotent group is a finite group in which every Sylow subgroup is normal.

Related facts

Proof

Proof using subgroup property collapse

For any finite group, we have:

Sylow $\implies$ automorph-conjugate

And for an ACIC-group, we have:

Automorph-conjugate $\implies$ normal

Thus, for a finite ACIC-group, we have:

Sylow $\implies$ Normal

Which is precisely the condition for being a finite nilpotent group.