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 $\struct {\Q, +}$.
$\blacksquare$