The concept of generalized digroup was proposed by Salazar-Díaz, Velásquez and Wills-Toro in their paper “Generalized digroups” as a non trivial extension of groups. In this way, many concepts and results given in the category of groups can be extended in a natural form to the category of generalized digroups. The aim of this paper is to present the construction of the free generalized digroup and study its properties. Although this construction is vastly different from the one given for the case of groups, we will use this concept, the classical construction for groups and the semidirect product to construct the tensor generalized digroup as well as the semidirect product of generalized digroups. Additionally, we give a new structural result for generalized digroups using compatible actions of groups and an equivariant map from a group set to the group corresponding to notions of associative dialgebras and augmented racks.
Generalized digroups are considered a nontrivial extension of the concept of group, thus one might think that many definitions and results on group theory can be naturally extended to generalized digroups. In this paper, we prove that it is not always true, since we do not have a version of Lagrange’s theorem for generalized digroups. On the other side, we propose and study Sylow-type theorems for generalized digroups.
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