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2019 Lagrange's theorem for Hom-Groups
Mohammad Hassanzadeh
Rocky Mountain J. Math. 49(3): 773-787 (2019). DOI: 10.1216/RMJ-2019-49-3-773

Abstract

Hom-groups are nonassociative generalizations of groups where the unitality and associativity are twisted by a map. We show that a Hom-group $(G, \alpha )$ is a pointed idempotent quasigroup (pique). We use Cayley tables of quasigroups to introduce some examples of Hom-groups. Introducing the notions of Hom-subgroups and cosets we prove Lagrange's theorem for finite Hom-groups. This states that the order of any Hom-subgroup $H$ of a finite Hom-group $G$ divides the order of $G$. We linearize Hom-groups to obtain a class of nonassociative Hopf algebras called Hom-Hopf algebras. As an application of our results, we show that the dimension of a Hom-sub-Hopf algebra of the finite dimensional Hom-group Hopf algebra $\mathbb {K}G$ divides the order of $G$. The new tools introduced in this paper could potentially have applications in theories of quasigroups, nonassociative Hopf algebras, Hom-type objects, combinatorics, and cryptography.

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Mohammad Hassanzadeh. "Lagrange's theorem for Hom-Groups." Rocky Mountain J. Math. 49 (3) 773 - 787, 2019. https://doi.org/10.1216/RMJ-2019-49-3-773

Information

Published: 2019
First available in Project Euclid: 23 July 2019

zbMATH: 07088336
MathSciNet: MR3983300
Digital Object Identifier: 10.1216/RMJ-2019-49-3-773

Rights: Copyright © 2019 Rocky Mountain Mathematics Consortium

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Vol.49 • No. 3 • 2019
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