Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-26T17:38:31.473Z Has data issue: false hasContentIssue false
  • English
  • Français

Variations on results on orders of products in finite groups

Part of: Representation theory of groups

Published online by Cambridge University Press:  12 October 2020

Juan Martínez  and
Alexander Moretó
Juan Martínez
Affiliation:
Departament de Matemàtiques, Universitat de València, 46100 Burjassot, València, Spain ([email protected]; [email protected])
Alexander Moretó
Affiliation:
Departament de Matemàtiques, Universitat de València, 46100 Burjassot, València, Spain ([email protected]; [email protected])
Get access Rights & Permissions [Opens in a new window]

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.

Keywords

element ordersnilpotent groupnilpotent Hall subgroupnormal Hall subgroup

MSC classification

Primary: 20D99: None of the above, but in this section
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 5 , October 2021 , pp. 1443 - 1449
Check for updates
Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Baumslag, B. and Wiegold, J.. A sufficient condition for nilpotency in a finite group. arXiv:1411.2877v1.Google Scholar
Beltrán, A., Lyons, R., Moretó, A., Navarro, G., Sáez, A. and Tiep, P. H.. Order of products of elements in finite groups. J. London Math. Soc. 99 (2019), 535–552.CrossRefGoogle Scholar
Beltrán, A. and Sáez, A.. Existence of normal Hall subgroups by means of orders of products. Math. Nachr. 292 (2019), 720–723.CrossRefGoogle Scholar
Guralnick, R. M. and Tiep, P. H.. Lifting in Frattini covers and a characterization of finite solvable groups. J. fur die Reine und Angew. Math. 708 (2015), 49–72.Google Scholar
Itô, N.. Note on (LM)-groups of finite order. Kodai Math. Sem. Rep. 2 (1951), 1–6.Google Scholar
Lee, H.. Triples in Finite Groups and a Conjecture of Guralnick and Tiep. PhD thesis. University of Arizona, 2017.Google Scholar
Moretó, A.. Sylow numbers and nilpotent Hall subgroups. J. Algebra 379 (2013), 80–84.CrossRefGoogle Scholar
Moretó, A. and Sáez, A.. Prime divisors of orders of products. Proc. R. Soc. Edinburgh Sect. A. 149 (2019), 1153–1162.CrossRefGoogle Scholar
Robinson, D. J. S.. A course in the theory of groups, 2nd edn (New York: Springer-Verlag, Inc. 1996).CrossRefGoogle Scholar
Sáez, A.. Divisores primos del orden de productos de elementos en grupos finitos. PhD Thesis. Universitat de València, 2018.Google Scholar