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Exercise (Cyclic Groups Are Abelian)

Suppose that $G$ is a cyclic group. Prove that $G$ is an Abelian group.

Click to see a solution.

Since $G$ is cyclic, we know that there exists some $a\in G$ with $\langle a\rangle = G$. Let $x,y\in G$. We need to show that $xy=yx$. But we know that $x=a^m$ and $y=a^n$ for some $n,m\in \mathbb{Z}$ because $G$ is cyclic. This means that $$xy=a^ma^n=a^{m+n}=a^{n+m}=a^na^m=yx,$$ which is what we needed to show.

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Page last modified on October 23, 2013, at 12:23 PM

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HomePage Current Problems All Problems Pictures ProblemsByWeek List of Problems List of Definitions List of Theorems List of Conjectures All Recent changes (use this to find a page that may have been lost)	Exercise Cyclic Groups Are Abelian Please Login to access more options. Exercise (Cyclic Groups Are Abelian) Suppose that $G$ is a cyclic group. Prove that $G$ is an Abelian group. Click to see a solution. Since $G$ is cyclic, we know that there exists some $a\in G$ with $\langle a\rangle = G$. Let $x,y\in G$. We need to show that $xy=yx$. But we know that $x=a^m$ and $y=a^n$ for some $n,m\in \mathbb{Z}$ because $G$ is cyclic. This means that $$xy=a^ma^n=a^{m+n}=a^{n+m}=a^na^m=yx,$$ which is what we needed to show. Edit Page History Source Attach File Backlinks List Group Page last modified on October 23, 2013, at 12:23 PM
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