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.