Chapter 13 - p-Automorphisms with p fixed points

Summary

In the extreme case, where a p-automorphism of a finite p-group has only P fixed points, the result is extremely strong.

Theorem 13.1.If a finite p-group P admits an automorphism φ of order pⁿ with exactly P fixed points, then P has a subgroup of (p,n)-bounded index which is nilpotent of class at most 2 (abelian, if P = 2).

For |φ| = p this was proved by C. R. Leedham-Green and S. McKay [1976] and by R. Shepherd [1971]; in the general case it was proved by S. McKay [1987] and by I. Kiming [1988].

We give a proof which is different from the original ones; although with possibly worse bounds for the index of the subgroup, our proof is more Lie ring oriented, making use of Higman's and Kreknin's Theorems from Chapter 7, the theory of powerful p-groups from Chapter 11, and the Lazard Correspondence from Chapter 10. As in Chapters 8 and 12, bounds for the ranks of abelian sections allow us to assume P to be powerful. Using a generalization of Maschke's Theorem, one can show that every (φ-invariant abelian section is a kind of “almost one-dimensional” ℤ〈φ〉-module. This information is used in a reduction to the case where P is uniformly powerful, and later in the proof of a Lie ring theorem. An application of Higman's Theorem to a subring of the associated Lie ring L(P) allows us to assume P to be nilpotent of class h(p), the value of Higman's function.