Gyrogroups and the Cauchy property

A gyrogroup is a nonassociative group-like structure. In this article, we extend the Cauchy property from groups to gyrogroups. The (weak) Cauchy property for finite gyrogroups states that if p is a prime dividing the order of a gyrogroup G, then G contains an element of order p. An application of a...

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Main Authors: Suksumran T., Ungar A.
Format: Journal
Published: 2017
Online Access:https://www.scopus.com/inward/record.uri?partnerID=HzOxMe3b&scp=85028588726&origin=inward
http://cmuir.cmu.ac.th/jspui/handle/6653943832/42435
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Institution: Chiang Mai University
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spelling th-cmuir.6653943832-424352017-09-28T04:27:05Z Gyrogroups and the Cauchy property Suksumran T. Ungar A. A gyrogroup is a nonassociative group-like structure. In this article, we extend the Cauchy property from groups to gyrogroups. The (weak) Cauchy property for finite gyrogroups states that if p is a prime dividing the order of a gyrogroup G, then G contains an element of order p. An application of a result in loop theory shows that gyrogroups of odd order as well as solvable gyrogroups satisfy the Cauchy property. Although gyrogroups of even order need not satisfy the Cauchy property, we prove that every gyrogroup of even order contains an element of order two. As an application, we prove that every group of order nq, where n ⊂ N and q is a prime with n < q, contains a unique characteristic subgroup of order q. 2017-09-28T04:27:05Z 2017-09-28T04:27:05Z 2016-01-01 Journal 15612848 2-s2.0-85028588726 https://www.scopus.com/inward/record.uri?partnerID=HzOxMe3b&scp=85028588726&origin=inward http://cmuir.cmu.ac.th/jspui/handle/6653943832/42435
institution Chiang Mai University
building Chiang Mai University Library
country Thailand
collection CMU Intellectual Repository
description A gyrogroup is a nonassociative group-like structure. In this article, we extend the Cauchy property from groups to gyrogroups. The (weak) Cauchy property for finite gyrogroups states that if p is a prime dividing the order of a gyrogroup G, then G contains an element of order p. An application of a result in loop theory shows that gyrogroups of odd order as well as solvable gyrogroups satisfy the Cauchy property. Although gyrogroups of even order need not satisfy the Cauchy property, we prove that every gyrogroup of even order contains an element of order two. As an application, we prove that every group of order nq, where n ⊂ N and q is a prime with n < q, contains a unique characteristic subgroup of order q.
format Journal
author Suksumran T.
Ungar A.
spellingShingle Suksumran T.
Ungar A.
Gyrogroups and the Cauchy property
author_facet Suksumran T.
Ungar A.
author_sort Suksumran T.
title Gyrogroups and the Cauchy property
title_short Gyrogroups and the Cauchy property
title_full Gyrogroups and the Cauchy property
title_fullStr Gyrogroups and the Cauchy property
title_full_unstemmed Gyrogroups and the Cauchy property
title_sort gyrogroups and the cauchy property
publishDate 2017
url https://www.scopus.com/inward/record.uri?partnerID=HzOxMe3b&scp=85028588726&origin=inward
http://cmuir.cmu.ac.th/jspui/handle/6653943832/42435
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