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 $$L^{\mathrm{gyr}}(G) = \{f\in L(G)\colon \forall a, x, y, z\in G, f(a\oplus\mathrm{gyr}[x, y]z) = f(a\oplus z)\}$$ in a natural way, where $L(G)$ is the space of all functions from $G$ into a field.