Non-Zero Rational Numbers under Multiplication form Infinite Abelian Group

Theorem

Let $\Q_{\ne 0}$ be the set of non-zero rational numbers:

$\Q_{\ne 0} = \Q \setminus \set 0$

The structure $\struct {\Q_{\ne 0}, \times}$ is a countably infinite abelian group.

Proof

From the definition of rational numbers, the structure $\struct {\Q, + \times}$ is constructed as the field of quotients of the integral domain $\struct {\Z, +, \times}$ of integers.

Hence from Multiplicative Group of Field is Abelian Group, $\struct {\Q_{\ne 0}, \times}$ is an abelian group.

From Rational Numbers are Countably Infinite, we have that $\struct {\Q_{\ne 0}, \times}$ is a countably infinite group.

$\blacksquare$

Also see

• Strictly Positive Rational Numbers under Multiplication form Countably Infinite Abelian Group

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 7$: Semigroups and Groups: Example $7.2$
• 1968: Ian D. Macdonald: The Theory of Groups ... (previous) ... (next): $\S 1$: Some examples of groups: Example $1.05$
• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: The Definition of Group Structure: $\S 29 \alpha \ (2)$
• 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 $\text{(ii)}$
• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $1$: Definitions and Examples: Example $1.5$

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• 1992: William A. Adkins and Steven H. Weintraub: Algebra: An Approach via Module Theory ... (previous) ... (next): $\S 1.1$: Example $2$