Invariant special subgroup lemma

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Statement

Suppose P is a finite p-group, i.e., a group of prime power order. Suppose A is a subgroup of \operatorname{Aut}(P) such that p does not divide the order of A. Suppose \varphi is a non-identity element of A. Then, there exists a subgroup Q of P that is either special or elementary Abelian, with the following properties:

  • \varphi acts nontrivially on Q/\Phi(Q).
  • A acts irreducible on Q/\Phi(Q).
  • \varphi acts trivially on \Phi(Q).

Facts used

  1. Structure lemma for p-group with coprime automorphism group having automorphism trivial on invariant subgroups

Proof

Let Q be minimal among the nontrivial A-invariant subgroups of P on which \varphi acts nontrivially. (Note that this collection of subgroups is nonempty, since \varphi acts nontrivially on P). Fact (1) then yields that Q satisfies the required conditions.

References

Textbook references