Cyclic Group is Abelian/Proof 1

Theorem

Let $G$ be a cyclic group.

Then $G$ is abelian.

Proof

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:

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

Thus:

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

and $G$ is by definition abelian.

$\blacksquare$

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 5.4$. Cyclic groups
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 39.3$ Cyclic Groups
• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $4$: Subgroups: Proposition $4.12$