Extension of Maschke's theorem

TL;DR

This paper extends Maschke's theorem to finite gyrogroups, establishing conditions for their linear representations and exploring the structure of their regular actions on specific function spaces.

Contribution

It proves Maschke's theorem for gyrogroups and examines their regular representations on a specialized function space, bridging gyrogroup theory with classical representation results.

Findings

Maschke's theorem holds for finite gyrogroups.

Characterization of gyrogroup representations analogous to groups.

Introduction of a new function space for gyrogroup actions.

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