Brauer core

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 defining ingredient::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.

Group properties

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

Subgroup properties

 * Characteristic subgroup

Subgroup-defining function properties
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.

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.