Groups having all elements off a normal subgroup with prime power order

Abstract

We consider a finite group $G$ with a normal subgroup $N$ so that all elements of $G\setminus N$ have prime power order. We prove that if there is a prime $p$ so that all the elements in $G\setminus N$ have $p$-power order, then either $G$ is a $p$-group or $G=PN$ where $P$ is a Sylow $p$-subgroup and $(G,P,P\cap N)$ is a Frobenius-Wielandt triple. We also prove that if all the elements of $G\setminus N$ 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 $G\setminus N$ 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.

15 pages