  In this paper we present some algebraic properties of subgroupoids and normal
subgroupoids. We define the normalizer of a wide subgroupoid $\mathcal{H}$ and
show that, as in the case of groups, the normalizer is the greatest wide
subgroupoid of the groupoid $\mathcal{G}$ in which $\mathcal{H}$ is normal.
Furthermore, we give the definition of center and commutator and prove that
both are normal subgroupoids, the first one of the union of all the isotropy
groups of $\mathcal{G}$ and the second one of $\mathcal{G}$. Finally, we
introduce the concept of inner isomorphism of $\mathcal{G}$ and show that the
set of all the inner isomorphisms of $\mathcal{G}$ is a normal subgroupoid,
which is isomorphic to the quotient groupoid of $\mathcal{G}$ by its center
$\mathcal{Z}(\mathcal{G})$, which extends to groupoids a well-known result in
groups.
