📂Abstract Algebra

Prove that All Cyclic groups are Abelian

Theorem ¹

All cyclic groups are abelian.

Explanation

It is also a fact that follows naturally without needing separate proof, if one shows that cyclic groups are isomorphic to the group of integers.

Proof

For a cyclic group $G := \left< a \right>$, let $g_{1} = a^{r}$ and $g_{2} = a^{s}$. $$ g_{1} g_{2} = a^{r} a^{s} = a^{r+s} = a^{s+r} = a^{s} a^{r} = g_{2} g_{1} $$ therefore, $G$ is an abelian group.

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