Chapter 8 - Almost regular automorphism of order p: almost nilpotency of p-bounded class

Summary

Now we are in a position to prove the first of the main theorems on finite p-groups with p-automorphisms having few fixed points. If such an automorphism is “almost regular”, with p^m fixed points, then the group is “almost nilpotent”: it has a subgroup of (p, m)-bounded index and of nilpotency class at most h(p), where h is Higman's function. This bound for the nilpotency class of a subgroup of (p, m)-bounded index is best possible, if required to depend on the order of the automorphism only. The result of this chapter will be used in Chapters 13 and 14.

Theorem 8.1.If a finite p-group P admits an automorphism φ of prime order p with exactly p^m fixed points, then P has a characteristic subgroup of (p,m)-bounded index which is nilpotent of class at most h(p), where h(p) is the value of Higman's function.

The proof relies on Higman's Theorem from § 7.2 on regular automorphisms of Lie rings in its combinatorial form and on the use of the associated Lie rings. Note that, at a first glance, an application of Higman's Theorem to L(P) and the induced automorphism φ cannot give us much information, since not only is φ not regular, but the number of fixed points of φ on L(P) can be much greater than on P, by a factor equal to the nilpotency class, say. Anotherimportant tool in the proof is a theorem of P. Hall from § 4.2.