What's the difference between a monoid and a group? What's the difference of a monoid and a group? I'm reading this book and it says that a group is a monoid with invertibility and this property is made to solve the equation $x \ast m=e$ and $m \ast x=e$ for $x$, where $m$ is any element of the structure.
I got confused because it's similar to the monoid's commutativity property which says that $m \ast n=n*m$ for all $m, n \in M$.

Elements of a monoid do not necessarily have inverse elements, while those of a group do. See 
http://en.wikipedia.org/wiki/Monoid
There are 4 axioms that define a group, one of which is the presence of inverse elements. Monoids only need to satisfy the other 3.
