Lagrange's Theorem For Hom-Groups

TL;DR

This paper extends Lagrange's theorem to finite Hom-groups, a nonassociative algebraic structure, and explores their linearization into Hom-Hopf algebras, with implications for various mathematical and cryptographic fields.

Contribution

It proves Lagrange's theorem for finite Hom-groups and introduces Hom-Hopf algebras, expanding the understanding of nonassociative algebraic structures.

Findings

Lagrange's theorem holds for finite Hom-groups.

Hom-groups can be linearized into Hom-Hopf algebras.

The dimension of Hom-sub-Hopf algebras divides the order of the original Hom-group.

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 table 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 KG 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.