Additive Group of Integers is Subgroup of Rationals
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Theorem
Let $\struct {\Z, +}$ be the additive group of integers.
Let $\struct {\Q, +}$ be the additive group of rational numbers.
Then $\struct {\Z, +}$ is a subgroup of $\struct {\Q, +}$.
Proof
Recall that Integers form Integral Domain.
The set $\Q$ of rational numbers is defined as the field of quotients of the integers.
The fact that the integers are a subgroup of the rationals follows from the work done in proving the Existence of Field of Quotients from an integral domain.
$\blacksquare$
Sources
- 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 $4$: Subgroups: Example $4.3$