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

Definition (abelian):

Let $G$ be a group. We call $G$ an abelian group if and only if for all $g,h\in G$, we have $gh=hg$ (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. $G=\langle g\rangle$ for a certain $g\in G$.

Proposition (cyclic group is abelian):

Let $G$ be a cyclic group. Then $G$ is abelian.

Proof: Indeed, write any two elements $h,j\in G$ as $h=g^{n}$, $j=g^{m}$, where $g\in G$ is such that $G=\langle g\rangle$. Then $hg=g^{n}g^{m}=g^{m}g^{n}=gh$, using associativity. $\Box$