📂Abstract Algebra

Proof That Every Cyclic Group Is Abelian

Theorem ¹

Every cyclic group is Abelian.

Explanation

Even without a separate proof, this is a fact that follows naturally once we show that a cyclic group is 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 Abelian.

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