On the p -regular G-conjugacy classes with sizes 1 or minimal

Abstract

Let N be a p-solvable normal subgroup of a finite group G, and N_p′ be a p^′ -Hall subgroup of N . If x is an element of G, then the conjugacy class size |x^G| = |G : CG(x)|. Obviously, when CG(x) is a maximal subgroup of G, |x^G| is minimal. In this paper, we investigate the structure of N_p′ by assuming that |x^G | is 1 or minimal for every p-regular element x of N