Cyclic Group is Abelian/Proof 1
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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 $4.12$