Extension of Maschke's theorem

Abstract

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

17 pages