Abstract

AVILA, Jesús  and  MARIN, Víctor. The Notions of Center, Commutator and Inner Isomorphism for Groupoids. ing.cienc. [online]. 2020, vol.16, n.31, pp.7-26. ISSN 1794-9165.  https://doi.org/10.17230/ingciencia.16.31.1.

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.

Keywords : Groupoid; normal subgroupoid; normalizer; center; commutator and inner isomorphisms.

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