What's the difference of a monoid and a group? I'm reading a book "Rubato music composer" and it says that a group is a monoid, but with invertibility and this property is made to solve the desire to solve the equation x*m=e and m*x=e for x, where m is any is any element of the structure.
I got confused because it's similar to the monoid's commutativity propertym which says that m*n=n*m for all m,n E M.