  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.
