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Using an algebraic point of view we present an introduction to the groupoidtheory, that is, we give fundamental properties of groupoids as, uniqueness ofinverses and properties of the identities, and study subgroupoids, widesubgroupoids and normal subgroupoids. We also present the isomorphism theoremsfor groupoids and as an applications we obtain the corresponding version ofZassenhaus Lemma and Jordan-H\"{o}lder Theorem for groupoids. Finally inspiredby the Ehresmann-Schein-Nambooripad Theorem we improve a result of R. Exelconcerning a one to one correspondence between partial actions of groups andactions of inverse semigroups.

Isomorphism Theorems for Groupoids and Some Applications