Additive Group of Integers is Countably Infinite Abelian Group

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Theorem

The set of integers under addition $\struct {\Z, +}$ forms a countably infinite abelian group.


Proof

From Integers under Addition form Abelian Group, $\struct {\Z, +}$ is an abelian group.

From Integers are Countably Infinite, the set of integers can be placed in one-to-one correspondence with the set of natural numbers.

Hence by definition, the underlying set of $\struct {\Z, +}$ is countably infinite.

$\blacksquare$


Sources