Class sizes of prime-power orden p'-elements and normal subgroups

Abstract

We prove an extension of the renowned Itô’s theorem on groups having two class sizes in three different directions at the same time: normal subgroups, p' -elements and prime- power order elements. Let N be a normal subgroup of a finite group G and let p be a fixed prime. Suppose that | x G |= 1or m for every q -element of N and for every prime q = p . Then, N has nilpotent p -complements.