Variations on results on orders of products in finite groups. Issue 5 (October 2021)
- Record Type:
- Journal Article
- Title:
- Variations on results on orders of products in finite groups. Issue 5 (October 2021)
- Main Title:
- Variations on results on orders of products in finite groups
- Authors:
- Martínez, Juan
Moretó, Alexander - 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.
- Is Part Of:
- Proceedings. Volume 151:Issue 5(2021)
- Journal:
- Proceedings
- Issue:
- Volume 151:Issue 5(2021)
- Issue Display:
- Volume 151, Issue 5 (2021)
- Year:
- 2021
- Volume:
- 151
- Issue:
- 5
- Issue Sort Value:
- 2021-0151-0005-0000
- 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 - Journal URLs:
- http://journals.cambridge.org/action/displayJournal?jid=PRM ↗
- DOI:
- 10.1017/prm.2020.66 ↗
- Languages:
- English
- ISSNs:
- 0308-2105
- Deposit Type:
- Legaldeposit
- View Content:
- Available online (eLD content is only available in our Reading Rooms) ↗
- Physical Locations:
- British Library HMNTS - ELD Digital store
- Ingest File:
- 19058.xml