Cyclic Group is Abelian/Proof 2

Theorem

Let $G$ be a cyclic group.


Then $G$ is abelian.


Proof

We have that Integers under Addition form Abelian Group.

The result then follows from combining:

Epimorphism from Integers to Cyclic Group
Epimorphism Preserves Commutativity.

$\blacksquare$


Sources

  • 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $25$. Cyclic Groups and Lagrange's Theorem: Theorem $25.3$
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