Let be a group and denote by the Frattini subgroup of . Prove that is characteristic.
We begin with some lemmas. Let denote the set of subgroups of ; recall that acts on by application: .
Lemma 1: Let be a group. If is an orbit of the action of on , then is characteristic in . Proof: Let be an automorphism of . Now , since permutes the elements of .
Lemma 2: The set of all maximal subgroups of is a union of orbits under the action of . Proof: It suffices to show that if and are maximal and nonmaximal subgroups of , respectively, then no automorphism carries to . Suppose to the contrary that is such an automorphism and that . Then , a contradiction.
Now to the main result. We see that is an intersection of characteristic subgroups, hence characteristic.