Invariant special subgroup lemma
From Groupprops
Statement
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 .
Facts used
Proof
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.
References
Textbook references
- Finite Groups by Daniel Gorenstein, ISBN 0821843427, ^{More info}, Page 183, Theorem 3.8, Section 5.3 (-automorphisms of -groups)