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

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Statement

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

Proof

Given: A group G, a prime p dividing the order of G such that O_{p'}(G)N_G(Z(J(P)) = G.

To prove: G is not simple non-abelian.

Proof: We consider two cases:

  • O_{p'}(G) is nontrivial: In this case, O_{p'}(G) is a proper nontrivial normal subgroup of G. It is proper because p divides the order of G.
  • O_{p'}(G) is trivial: In this case, N_G(Z(J(P))) = G. Thus, Z(J(P)) is normal in G. Since p divides the order of G, P is nontrivial, hence, since the ZJ-functor, by definition, sends nontrivial p-subgroups to nontrivial p-subgroups, Z(J(P)) is nontrivial. Thus, Z(J(P)) is a nontrivial normal subgroup of G. The only way it can be the whole group is if P is abelian. Hence, we either have a proper nontrivial normal subgroup or have that P is abelian. In either case, we are done.