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, α) 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 Homgroups 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.

Hom-Lie Algebras and Hom-Lie Groups, Integration and Differentiation

“…Throughout this paper, we consider regular Hom-groups that is the case when the structure map is invertible, and this notion can be traced back to Caenepeel and Goyvaerts's pioneering work [3]. The axioms in the following definition of Hom-group is different from the one in [7,8,13]. However, we show that if the structure map is invertible, then some axioms in the original definition are redundant and can be obtained from the Hom-associativity condition.…”

“…Remark 2.11. Note that the definition of a Hom-group in [8] consists of the axiom Φ(e Φ ) = e Φ . In Proposition 2.13, we show that this axiom is redundant in the regular case.…”

“…Recently, M. Hassanzadeh developed representations and a (co)homology theory for Hom-groups in [7]. He also proved Lagrange's theorem for finite Hom-groups in [8]. The recent developments on Hom-groups (see [7,8,13]) make it natural to study Hom-Lie groups and to explore the relationship between Hom-Lie groups and Hom-Lie algebras.…”

Free Hom-groups, Hom-rings and Semisimple modules

“…Recently, M. Hassanzadeh developed representations and a (co)homology theory for Hom-groups in [7]. He also proved Lagrange's theorem for finite Hom-groups in [8]. In [9], Hom-Lie groups and relationship between Hom-Lie groups and Hom-Lie algebras were explored.…”

Matching Hom-Setting of Rota-Baxter Algebras, Dendriform Algebras, and Pre-Lie Algebras

“…It is shown that a Homassociative algebra gives rise to a Hom-Lie algebra using the commutator. Since then, various Hom-analogues of some classical algebraic structures have been introduced and studied intensively, such as Hom-coalgebras, Hom-bialgebras and Hom-Hopf algebras [24,25], Hom-groups [26,27], Hom-Hopf modules [28], Hom-Lie superalgebras [29,30], generalize Hom-Lie algebras [31], and Hom-Poisson algebras [32].…”