# Nilpotent not implies ACIC

This article gives the statement and possibly, proof, of a non-implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., nilpotent group) neednotsatisfy the second group property (i.e., ACIC-group)

View a complete list of group property non-implications | View a complete list of group property implications

Get more facts about nilpotent group|Get more facts about ACIC-group

This article gives the statement and possibly, proof, of a non-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., automorph-conjugate subgroup) neednotsatisfy the second subgroup property (i.e., characteristic subgroup)

View all subgroup property non-implications | View all subgroup property implications

## Contents

## Statement

A nilpotent group need not be ACIC: in a nilpotent group, every automorph-conjugate subgroup need not be characteristic.

## Proof

### Example of a non-Abelian group of prime-cubed order

`Further information: prime-cube order group:p2byp`

Let be an odd prime. Consider the group of order obtained as a semidirect product of a cyclic group of order with a cyclic group of order . In other words, is given by:

Then, the cyclic group of order generated by is an automorph-conjugate subgroup that is not characteristic.

is clearly not characteristic because it is not normal, as it is the non-normal part in a semidirect product.

To see that it is automorph-conjugate, observe that any automorphism must send it to another cyclic group of order . Now, all cyclic groups of order lie inside the subgroup , which is elementary Abelian of order generated by and . There are exactly subgroups inside this of order , and one of them, the center (generated by ) is characteristic. Of the other subgroups, none is normal in , and hence, since the size of any conjugacy class of subgroups is a multiple of , they are all conjugate. Thus, is an automorph-conjugate subgroup.

### Example of non-Abelian groups of prime-fourth order

**PLACEHOLDER FOR INFORMATION TO BE FILLED IN**: [SHOW MORE]