# Brauer core

This article defines a subgroup-defining function, viz., a rule that takes a group and outputs a unique subgroupTemplate:Group property core operator

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

## Contents

## Definition

### Symbol-free definition

The **Brauer core** of a finite group is defined as the unique largest normal subgroup of odd order.

### Definition with symbols

The **Brauer core** of a group , denoted as , is defined as the unique largest normal subgroup of among those of odd order. Equivalently, it is the pi-core of where is the set of all odd primes.

### In terms of the group property core operator

The Brauer core is a subgroup-defining function obtained by applying the group property core operator to the group property of being an odd-order group. The justification for applying this operator is the fact that the property of having odd order is a normal join-closed group property.

## Group properties

- Odd-order group
- Solvable group: This is on acount of its being of odd order

## Subgroup properties

## Subgroup-defining function properties

### Idempotence

This subgroup-defining function is idempotent. In other words, applying this twice to a given group has the same effect as applying it once

The Brauer core of the Brauer core is the Brauer core. In fact, a group equals its own Brauer core if and only if it has odd order.

### Quotient-idempotence

This subgroup-defining function is quotient-idempotent: taking the quotient of any group by the subgroup, gives a group where the subgroup-defining function yields the trivial subgroup

View a complete list of such subgroup-defining functions

The quotient of a group by its Brauer core has trivial Brauer core. In other words, the quotient map by the Brauer core is an idempotent quotient-defining function.

## Associated constructions

### Associated quotient-defining function

The quotient-defining function associated with this subgroup-defining function is:[[Brauer quotient]]