Is an abelian group closed?

Could you please clarify for me if an abelian group is indeed closed under its operation? I understand that an abelian group satisfies the properties of associativity, identity, inverses, and commutativity, but I'm specifically wondering about its closure. Does the fact that it is a group inherently mean that it is closed, or is there something specific about an abelian group that guarantees closure? I would appreciate your insight on this matter.