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

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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.