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

TL;DR

This paper characterizes the structure of finite groups with a normal subgroup where all elements outside have prime power order, revealing conditions under which the group is a p-group, Frobenius group, or nonsolvable.

Contribution

It provides new structural results for finite groups with elements outside a normal subgroup having prime power orders, including classifications involving Frobenius and Wielandt triples.

Findings

If all elements outside N have p-power order, G is either a p-group or a Frobenius-Wielandt extension.

If outside elements have orders divisible by two primes, G/N is a Frobenius or 2-Frobenius group.

With three or more primes dividing element orders outside N, G/N is nonsolvable.

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.