Groups Having all Elements off a Normal Subgroup with Prime Power Order

Abstract

We prove that if G is a finite group, N is a normal subgroup, and there is a prime p so that all the elements in G ∖ N have p-power order, then either G is a p-group or G = PN where P is a Sylow p-subgroup and (G,P,P ∩ N) is a Frobenius–Wielandt triple. We also prove that if all the elements of G ∖ N have prime power orders and the orders are divisible by two primes p and q, then G is a {p,q}-group and G/N is either a Frobenius group or a 2-Frobenius group. If all the elements of G ∖ N have prime power orders and the orders are divisible by at least three primes, then all elements of G have prime power order and G/N is nonsolvable.