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

2022/03/04 by Mark L. Lewis, Lewis, Mark L. · 1 citation
Mathematics · #20D99 #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #math.GR #msc:20D99

paper · pdf · doi:10.48550/arxiv.2203.02537

15 pages

arxiv created 2022/03/04 · openalex publication_date 2022/03/04 · arxiv updated 2022/03/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider a finite group G with a normal subgroup N so that all elements of G ∖ N have prime power order. We prove that if 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.

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