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 $1.4$

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