On the degree of commutativity of p -groups of maximal class

Abstract

Let G be a p -group of maximal class of order pm , p ≠ 2, and c (G) the degree of commutativity of G. Let c0 be the nonnegative residue of c modulo p – 1. In this paper, by using only Lie algebra techniques, we prove that 2c ≥ m – 2p + c 0 + 1. Also, we give examples of Lie algebras satisfying the following equalities: In addition, there exist examples of p -groups of maximal class satisfying 2c = m – 2p + c0 + 3 for each c0 ∈ [2, p – 2] (see [6, Theorem 4.5]). (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)