Cyclic Group is Abelian

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Let $G$ be a cyclic group.

Then $G$ is abelian.

Proof 1

Let $G$ be a cyclic group.

All elements of $G$ are of the form $a^n$, where $n \in \Z$.

Let $x, y \in G: x = a^p, y = a^q$.

From Powers of Group Elements: Sum of Indices:

$x y = a^p a^q = a^{p + q} = a^{q + p} = a^q a^p = y x$


$\forall x, y \in G: x y = y x$

and $G$ is by definition abelian.


Proof 2

We have that Integers under Addition form Abelian Group.

The result then follows from combining:

Epimorphism from Integers to Cyclic Group
Epimorphism Preserves Commutativity.