Extension of Maschke’s theorem

In the present article, we examine linear representations of finite gyrogroups, following their group-counterparts. In particular, we prove Maschke’s theorem for gyrogroups, along with its converse. This suggests studying the left regular action of a gyrogroup (G,⊕) on the function space Lgyr(G)={f∈L(G):∀a,x,y,z∈G,f(a⊕gyr[x,y]z)=f(a⊕z)} in a natural way, where L(G) is the space of all functions from G to a field.