# 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 quotient field of the integers.

The fact that the integers are a subgroup of the rationals follows from the work done in proving the Existence of Quotient Field 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 $4.3$