Group Theory/Abelian groups and the Grothendieck group of a monoid

abelian Let be a group. We call an abelian group if and only if for all , we have (where we denote the group operation by juxtaposition). cyclic group A cyclic group is a group that is generated by a single of its elements, ie. for a certain . cyclic group is abelian Let be a cyclic group. Then is abelian. Indeed, write any two elements as , , where is such that . Then , using associativity.