Infinite Cyclic Group is Unique up to Isomorphism

From ProofWiki
Jump to navigation Jump to search

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.