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

**Definition (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).

**Definition (cyclic group)**:

A **cyclic group** is a group that is generated by a single of its elements, ie. for a certain .

**Proposition (cyclic group is abelian)**:

Let be a cyclic group. Then is abelian.

**Proof:** Indeed, write any two elements as , , where is such that . Then , using associativity.