# Additive Group of Integers is Countably Infinite Abelian Group

## 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

- 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $2$: Subgroups and Cosets: $\S 38$ - 1996: John F. Humphreys:
*A Course in Group Theory*... (previous) ... (next): Chapter $1$: Definitions and Examples: Example $1.4$

- 1992: William A. Adkins and Steven H. Weintraub:
*Algebra: An Approach via Module Theory*... (previous) ... (next): $\S 1.1$: Example $1$