Every Cyclic Group is Abelian

Abelian Group problems and solutions

Problem 619

Prove that every cyclic group is abelian.

 
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Proof.

Let $G$ be a cyclic group with a generator $g\in G$.
Namely, we have $G=\langle g \rangle$ (every element in $G$ is some power of $g$.)

Let $a$ and $b$ be arbitrary elements in $G$.
Then there exists $n, m\in \Z$ such that $a=g^n$ and $b=g^m$.

It follows that
\begin{align*}
ab&=g^ng^m=g^{n+m}=g^mg^n=ba.
\end{align*}

Hence we obtain $ab=ba$ for arbitrary $a, b\in G$.
Thus $G$ is an abelian group.


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