Variations on results on orders of products in finite groups. Issue 5 (October 2021)

• Record Type: Journal Article
• Authors: Martínez, Juan; Moretó, Alexander
• Is Part Of: Proceedings. Volume 151:Issue 5(2021)
• Page Start: 1443
• Page End: 1449
• Publication Date: 2021-10
• Subjects: element orders -- nilpotent group -- nilpotent Hall subgroup -- normal Hall subgroup; 20D99; Mathematics -- Periodicals; 510
• DOI: 10.1017/prm.2020.66
• Languages: English
• ISSNs: 0308-2105

Abstract

Abstract : In 2014, Baumslag and Wiegold proved that a finite group G is nilpotent if and only if o ( xy ) = o ( x ) o ( y ) for every x, y ∈ G with ( o ( x ), o ( y )) = 1. This has led to a number of results that characterize the nilpotence of a group (or the existence of nilpotent Hall subgroups, or the existence of normal Hall subgroups) in terms of prime divisors of element orders. Here, we look at these results with a new twist. The first of our main results asserts that G is nilpotent if and only if o ( xy ) ⩽ o ( x ) o ( y ) for every x, y ∈ G of prime power order with ( o ( x ), o ( y )) = 1. As an immediate consequence, we recover the Baumslag–Wiegold theorem. The proof of this result is elementary. We prove some variations of this result that depend on the classification of finite simple groups.