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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 (juanma23@alumni.uv.es; Alexander.Moreto@uv.es)
Alexander Moretó
Affiliation:
Departament de Matemàtiques, Universitat de València, 46100 Burjassot, València, Spain (juanma23@alumni.uv.es; Alexander.Moreto@uv.es)
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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
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Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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References

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