Proof that cyclic groups are abelian

Proof that cyclic groups are abelian
Tag	`8MADRXG8PQ`
URI	`m2://8MADRXG8PQ`
Kind	proof
Writing	W0 stub
MSC	20A05
Proves	Cyclic groups are abelian
Uses	Cyclic group, Laws of exponents in a group, Abelian group
Methods	Direct proof

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Proof

Let $G$ be a cyclic group with generator $g$ .
Let $x$ and $y$ be elements of $G$ . Then there are integers $m$ and $n$ with $x = g^m$ and $y = g^n$ .
By the laws of exponents, $xy = g^m g^n = g^{m+n}$ .
In the same way, $yx = g^n g^m = g^{n+m}$ .
Since $m + n = n + m$ , $xy = yx$ . Thus $G$ is abelian.

Idea

Every element is a power of one generator, and powers of the same element commute.

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Binary operation definition
Group concept
Commutativity definition
Laws of exponents in a group statement
Subgroup definition
Abelian group definition
Cyclic subgroup generated by an element definition
Cyclic group definition
Proof that cyclic groups are abelian proof

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Cyclic groups are abelian 04RG02S6BA

Sources

Web https://en.wikipedia.org/wiki/Group_(mathematics), Cyclic groups

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