Comment by JuniperMesos

A group is an algebraic structure, and many algebraic structures are defined as sets with an operation that obeys some number of axioms. It happens that a group is a structure whose operation has exactly three axioms; but a monoid for instance is a set with an operation with just two axioms - it's a group without the "has inverses" axiom - and I don't think this makes the monoid worse than a group in any meaningful way.