# Additive Group of Integers is Normal Subgroup of Reals

Jump to navigation
Jump to search

## Theorem

Let $\struct {\Z, +}$ be the additive group of integers.

Let $\struct {\R, +}$ be the additive group of real numbers.

Then $\struct {\Z, +}$ is a normal subgroup of $\struct {\R, +}$.

## Proof

From Additive Group of Integers is Subgroup of Reals, $\struct {\Z, +}$ is a normal subgroup of $\struct {\R, +}$.

As the additive group of real numbers is abelian, from Subgroup of Abelian Group is Normal it follows that $\struct {\Z, +} \lhd \struct {\R, +}$.

$\blacksquare$

## Sources

- 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): Chapter $6$: Cosets: Exercise $14$