# Intermediately automorph-conjugate implies intermediately characteristic in nilpotent

From Groupprops

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties, when the big group is a nilpotent group. That is, it states that in a Nilpotent group (?), every subgroup satisfying the first subgroup property (i.e., Intermediately automorph-conjugate subgroup (?)) must also satisfy the second subgroup property (i.e., Intermediately characteristic subgroup (?)). In other words, every intermediately automorph-conjugate subgroup of nilpotent group is a intermediately characteristic subgroup of nilpotent group.

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## Contents

## Statement

In a nilpotent group, any intermediately automorph-conjugate subgroup is intermediately characteristic.

## Related facts

### Converse

The converse is clearly true in all groups: intermediately characteristic implies intermediately automorph-conjugate. In fact, in any group, characteristic implies automorph-conjugate.

### Removing the *intermediately* operator

It is *not* in general true in a nilpotent group that every automorph-conjugate subgroup is characteristic. In fact, a group satisfying the property that every automorph-conjugate subgroup is characteristic is termed an ACIC-group. Some facts:

## Facts used

- Intermediately automorph-conjugate implies intermediately normal-to-characteristic
- Intermediately normal-to-characteristic implies intermediately characteristic in nilpotent

## Proof

The proof follows directly by combining facts (1) and (2).