Additive Group of Integers is Subgroup of Rationals

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 $4.3$
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