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Homeproof that all cyclic groups are abelian

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# proof that all cyclic groups are abelian

The following is a proof that all cyclic groups are abelian.

###### Proof.

Let $G$ be a cyclic group and $g$ be a generator of $G$. Let $a,b\in G$. Then there exist $x,y\in{\mathbb{Z}}$ such that $a=g^{x}$ and $b=g^{y}$. Since $ab=g^{x}g^{y}=g^{{x+y}}=g^{{y+x}}=g^{y}g^{x}=ba$, it follows that $G$ is abelian. ∎

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## Mathematics Subject Classification

20A05*no label found*

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