A Generalized Frattini Subgroup of a Finite Group

Abstract

For a finite group G and an arbitrary prime p, let Sp(G) denote the intersection of all maximal subgroups M of G such that [G:M] is both composite and not divisible by p; if no such M exists we set Sp(G) = G. Some properties of G are considered involving Sp(G). In particular, we obtain a characterization of G when each M in the definition of Sp(G) is nilpotent. © 1989, Hindawi Publishing Corporation. All right reserved.