# Additive Group of Integers is Normal 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 normal subgroup of $\struct {\Q, +}$.

## Proof

From Additive Group of Integers is Subgroup of Rationals, $\struct {\Z, +}$ is a subgroup of $\struct {\Q, +}$.

From Rational Numbers under Addition form Infinite Abelian Group, $\struct {\Q, +}$ is an abelian group.

From Subgroup of Abelian Group is Normal it follows that $\struct {\Z, +}$ is a normal subgroup of $\struct {\Q, +}$.

$\blacksquare$