Certain relations between <i>p</i>-regular class sizes and the <i>p</i>-structure of <i>p</i>-solvable groups

Abstract

Abstract Let G be a finite p -solvable group for a fixed prime p . We study how certain arithmetical conditions on the set of p -regular conjugacy class sizes of G influence the p -structure of G . In particular, the structure of the p -complements of G is described when this set is {1, m, n } for arbitrary coprime integers m, n &gt; 1. The structure of G is determined when the noncentral p -regular class lengths are consecutive numbers and when all of them are prime powers.