# Oliver subgroup

This article defines a subgroup-defining function, viz., a rule that takes a group and outputs a unique subgroup

View a complete list of subgroup-defining functions OR View a complete list of quotient-defining functions

## Contents

## Definition

### The main Oliver subgroup

Suppose is a prime number and is a finite p-group. The **Oliver subgroup** of , denoted (don't know what that letter actually is) is defined as the unique largest subgroup of such that there exists an ascending series of subgroups of :

such that:

- Each is a normal subgroup of .
- We have the condition that for all

where the indicates an iterated commutator of the form with occurring times.

Here, denotes the first omega subgroup.

### Other Oliver subgroups

Suppose is a prime number and is a finite p-group. The **Oliver subgroup** of , denoted (don't know what that letter actually is) is defined as the unique largest subgroup of such that there exists an ascending series of subgroups of :

such that:

- Each is a normal subgroup of .
- We have the condition that for all

where the indicates an iterated commutator of the form with occurring times.

Here, denotes the first omega subgroup.

### Relation between definitions

The main Oliver subgroup is the Oliver subgroup.