Monoids in which each element has an inverse deserve a special name:
Definition
A group is a structure (G, *, e, x->x-1) consisting of a set G, a binary associative multiplication * with unit element e and a unary operation x -> x-1 such that x-1 is an inverse of x.
Notice that if G is a group, then (G,*,e) is indeed a monoid. Since every element of a monoid has at most one inverse, we could also have defined a group as a monoid in which every element has an inverse.
| Most properties that we have derived for monoids so far, also hold for groups. Similarly, notation introduced for monoids will also apply to groups. For example, a group is called commutative (or abelian) if the corresponding monoid is commutative, i.e., the multiplication is commutative. |  |