# Brauer core

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
Template:Group property core operator

## 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 $G$, denoted as $O(G)$, is defined as the unique largest normal subgroup of $G$ among those of odd order. Equivalently, it is the pi-core of $G$ where $\pi$ 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.

## 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]]