A geometrical approach to g-digroups

Abstract

Representation group theory plays an important role in group theory, much more in geometry, on which we can determine, in many cases, what kind of geometry or geometries a manifold M supports by knowing a faithful representation of the fundamental group Π1(M) in the group Iso + (M) of preserving-orientations isometries of M. Motivated by this and by the definition of a g-digroup action given in [1], in this paper we present a study of such action by proving some interesting results that are well known in group actions, for example: the orbit stabilizer theorem, the class equation, Burnside’s formula and we relate g-digroup actions with trioids. We introduce the definition of representation of a g-digroup into dialge-bras and a strong connection with g-digroup action, like in the case of group. We present the g-digroups as an invariant of quotient orbifolds.