# Glauberman type for a prime divisor implies not simple non-abelian

From Groupprops

## Statement

Suppose is a finite group and is a prime number dividing the order of . Suppose further that is a group of Glauberman type for . Then, is not a simple non-abelian group.

## Proof

**Given**: A group , a prime dividing the order of such that .

**To prove**: is not simple non-abelian.

**Proof**: We consider two cases:

- is nontrivial: In this case, is a proper nontrivial normal subgroup of . It is proper because divides the order of .
- is trivial: In this case, . Thus, is normal in . Since divides the order of , is nontrivial, hence, since the ZJ-functor, by definition, sends nontrivial -subgroups to nontrivial -subgroups, is nontrivial. Thus, is a nontrivial normal subgroup of . The only way it can be the whole group is if is abelian. Hence, we either have a proper nontrivial normal subgroup or have that is abelian. In either case, we are done.