Invariant special subgroup lemma
Suppose is a finite -group, i.e., a group of prime power order. Suppose is a subgroup of such that does not divide the order of . Suppose is a non-identity element of . Then, there exists a subgroup of that is either special or elementary Abelian, with the following properties:
- acts nontrivially on .
- acts irreducible on .
- acts trivially on .
- Structure lemma for p-group with coprime automorphism group having automorphism trivial on invariant subgroups
Let be minimal among the nontrivial -invariant subgroups of on which acts nontrivially. (Note that this collection of subgroups is nonempty, since acts nontrivially on ). Fact (1) then yields that satisfies the required conditions.