Rational Numbers under Addition form Infinite Abelian Group
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Theorem
Let $\Q$ be the set of rational numbers.
The structure $\struct {\Q, +}$ is a countably infinite abelian group.
Proof
The rational numbers are, by definition, the field of quotients of the integral domain $\struct {\Z, +, \times}$ of integers.
Hence by definition, $\struct {\Q, +, \times}$ is a field.
The fact that $\struct {\Q, +}$ forms an abelian group follows directly from the definition of a field.
From Rational Numbers are Countably Infinite, we have that $\struct {\Q, +}$ is countably infinite.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 7$: Semigroups and Groups: Example $7.1$
- 1968: Ian D. Macdonald: The Theory of Groups ... (previous) ... (next): $\S 1$: Some examples of groups: Example $1.02$
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: The Definition of Group Structure: $\S 29 \alpha \ (1)$
- 1974: Thomas W. Hungerford: Algebra ... (previous) ... (next): $\text{I}$: Groups: $\S 1$: Semigroups, Monoids and Groups
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 34$. Examples of groups: $(1)$
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.2$: Groups; the axioms: Examples of groups $\text{(i)}$
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $1$: Definitions and Examples: Example $1.4$