A note on maximal subgroups and conjugacy classes of finite groups

Abstract

Given a finite group G, two elements are ≡m-related if they lie in exactly the same maximal subgroups of G. This equivalence relation was introduced by P. J. Cameron, A. Lucchini and C. M. Roney-Dougal to understand better the generated set for finite groups. We study the relation ≡m in a conjugacy class of G and determine sufficient conditions to ensure that such conjugacy class is contained in the Fitting subgroup F(G) of G. If G is soluble and x, y are two ≡m-related elements of G such that (|xG|, |yG|) = 1, we prove that they lie in F(G).