Infinite Cyclic Group is Unique up to Isomorphism
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Theorem
All infinite cyclic groups are isomorphic.
That is, up to isomorphism, there is only one infinite cyclic group.
Proof
Let $G_1$ and $G_2$ be infinite cyclic groups.
From Infinite Cyclic Group is Isomorphic to Integers we have:
- $G_1 \cong \struct {\Z, +} \cong G_2$
where $\struct {\Z, +}$ is the additive group of integers.
From Isomorphism is Equivalence Relation it follows that:
- $G_1 \cong G_2$
$\blacksquare$
Comment
Now that as we have, in a sense, defined an infinite cyclic group with reference to the additive group of integers that we painstakingly constructed in the definition of integers, it naturally follows that we should use $\struct {\Z, +}$ as an "archetypal" infinite cyclic group.