Theorem 8.1 states that if a finite p-group P admits an automorphism of order P with pm fixed points, then P has a subgroup of (p, m)-bounded index which is nilpotent of p-bounded class. In this chapter we prove that the nilpotency class of a subgroup of (p, m)bounded index can be bounded in terms of m only. The following theorem is due to Yu. Medvedev [1994a,b].

Theorem 14.1.If a finite p-group P admits an automorphism φ of prime order p with exactly pm fixed points, then P has a subgroup of (p, m)-bounded index which is nilpotent of m-bounded class.

Neither Theorem 8.1 nor Theorem 14.1 follows from the other: if P is much less than m, then Theorem 8.1 gives a better result; on the other hand, if m is much less than p, then Theorem 14.1 is better. Theorem 14.1 confirmed the conjecture from [E. I. Khukhro, 1985] (also [Kourovka Notebook, 1986, Problem 10.68]). This conjecture was prompted by the result of C. R. Leedham-Green and S. McKay [1976] and R. Shepherd [1971] on p-groups of maximal class, which amounts to the special case of Theorem 13.1 where |φ| = |Cp(φ)| = P implies that P has a subgroup of p-bounded index which is nilpotent of class 2.

The proof of Theorem 14.1 is essentially about Lie rings; we use many of the of techniques developed in Chapter 13, including the lifted Lie products from [Yu. Medvedev, 1994b].