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]]
