On p-groups of maximal class. III

Abstract

In this paper, we consider the family of the p-groups of maximal class and study some invariants related to the conjugacy classes of these groups. This subject was first taken up in [7] and [6], whose notation and terminology we maintain and give after this introduction. In [7], the invariants σG and k(G) are found for the p-groups of maximal class satisfying |G| ≤ p7. Here, apart from extending those results to the case when G has order p8 or p9, we consider the problem in all its generality and propose to determine their values for as large a family of p-groups of maximal class as possible. In this direction, we get some general results in the first paragraph, from which we want to point out the following ones.