Groupoids: Direct products, semidirect products and solvability

Abstract

We present some constructions of groupoids such as: direct product, semidirect product and give necessary and sufficient conditions for a groupoid to be embedded into a direct product of groupoids. Also, we establish necessary and sufficient conditions to determine when a semidirect product is direct. Finally the notion of solvable groupoid is introduced and studied, in particular it is shown that a finite groupoid \(\mathcal{G}\) is solvable if and only if its isotropy groups are.