Extension of Maschke's theorem

In the present article, we examine linear representations of finite gyrogroups, following their group-counterparts. In particular, we prove the celebrated theorem of Maschke for gyrogroups, along with its converse. This suggests studying the left regular action of a gyrogroup $(G,\oplus )$ on the function space $$L^{\mathrm{g}\mathrm{y}\mathrm{r}}(G)=\{f\in L(G)\text{\hspace{0.17em}⁣}:\mathrm{\forall }a,x,y,z\in G,f(a\oplus \mathrm{g}\mathrm{y}\mathrm{r}[x,y]z)=f(a\oplus z)\}$$ in a natural way, where $L(G)$ is the space of all functions from $G$ into a field.