Chapter 12 - Almost regular automorphism of order pⁿ: almost solubility of pⁿ-bounded derived length

Summary

The second of the main results on almost regular p-automorphisms of finite p-groups is a match to Kreknin's Theorem on regular automorphisms of Lie rings. If a finite p-group P admits an automorphism φ of order pⁿ with exactly p^m fixed points, then P contains a subgroup of (p, m, n)-bounded index which is soluble of (p, n)-bounded derived length (that is, of derived length bounded in terms of the order of the automorphism only). Kreknin's Theorem is used twice in the proof. First it is applied to the associated Lie ring L(P), in the case where P is uniformly powerful, to prove that P is an extension of a group of (p, m, n)-bounded nilpotency class by a group of (p, n)-bounded derived length (this already gives a “weak” bound, in terms of p, m and n, for the derived length of P in the general case). Then free nilpotent ℚ-powered groups and the Mal'cev Correspondence are used to derive a consequence of Kreknin's Theorem, with a kind of a “weak” conclusion that depends on the nilpotency class. Rather miraculously, a combination of two “weak” results yields the desired “strong” bound, in terms of pⁿ only, for the derived length of a subgroup of (p, m, n)-bounded index.

By Lemma 2.12 the number of fixed points of φ in all φ-invariant sections of P is at most p^m; by Corollary 2.7 all these sections have rank at most mpⁿ. This is why powerful p-groups appear naturally in the proofs.