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