## Rocky Mountain Journal of Mathematics

### Lagrange's theorem for Hom-Groups

#### 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.

#### Article information

Source
Rocky Mountain J. Math., Volume 49, Number 3 (2019), 773-787.

Dates
First available in Project Euclid: 23 July 2019

https://projecteuclid.org/euclid.rmjm/1563847233

Digital Object Identifier
doi:10.1216/RMJ-2019-49-3-773

Mathematical Reviews number (MathSciNet)
MR3983300

Zentralblatt MATH identifier
07088336

#### Citation

Hassanzadeh, Mohammad. Lagrange's theorem for Hom-Groups. Rocky Mountain J. Math. 49 (2019), no. 3, 773--787. doi:10.1216/RMJ-2019-49-3-773. https://projecteuclid.org/euclid.rmjm/1563847233

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