The Notions of Center, Commutator and Inner Isomorphism for Groupoids

Abstract

In this paper we introduce some algebraic properties of subgroupoids and normal subgroupoids. we define other things, we define the normalizer of a wide subgroupoid H of a groupoid G and show that, as in the case of groups, this normalizer is the greatest wide subgroupoid of G in which H is normal. Furthermore, we provide definitions of the center Z(G) and the commutator G' of the groupoid G and prove that both of them are normal subgroupoids. We give the notions of inner and partial isomorphism of G and show that the groupoid I(G) given by the set of all the inner isomorphisms of G is a normal subgroupoid of A(G), the set of all the partial isomorphisms of G. Moreover, we prove that I(G) is isomorphic to the quotient groupoid G/Z(G), which extends to groupoids the corresponding well-known result for groups.

Groupoids: Direct products, semidirect products and solvability

“…If G is a groupoid and ∅ ̸ = B ⊆ G, then the subgroupoid given in the Proposition 3.2 is called the subgroupoid generated by B and it will be denoted by ⟨B⟩. It can be proved that the set ⟨B⟩ is given by [3,Proposition 4.4] we know that [G, G] is a normal subgroupoid of G. We proceed with the next.…”

“…The main goal of this paper is to continue the works [3,4] and [5] by presenting new constructions of groupoids and study some structural properties of them. For this, after the introduction in section 1 we introduce the necessary background on groupoids.…”

“…[3, Proposition 2.6] Let G be a groupoid, {H i } i∈I a family of subgroupoids of G, and ∅ ̸ = B ⊆ G. Then:…”

Generalizations of Lagrange and Sylow Theorems for Groupoids

“…In [2] isomorphism theorems for groupoids were proved, such as results of normal and subnormal groupoid series. In addition, in [3] the notions of center, commutator and inner isomorphism for groupoids were presented.…”