  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.
