# Quotient Group of Integers by Multiples

## Theorem

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

Let $\struct {m \Z, +}$ be the additive group of integer multiples of $m$.

Let $\struct {\Z_m, +_m}$ be the additive group of integers modulo $m$.

Then the quotient group of $\struct {\Z, +}$ by $\struct {m \Z, +}$ is $\struct {\Z_m, +_m}$.

Thus:

$\index \Z {m \Z} = m$

## Proof

From Subgroups of Additive Group of Integers, $\struct {m \Z, +}$ is a subgroup of $\struct {\Z, +}$.

From Subgroup of Abelian Group is Normal, $\struct {m \Z, +}$ is normal in $\struct {\Z, +}$.

Therefore the quotient group $\dfrac {\struct {\Z, +} } {\struct {m \Z, +} }$ is defined.

Now $\Z$ modulo $m \Z$ is Congruence Modulo a Subgroup.

This is merely congruence modulo an integer.

Thus the quotient set $\Z / m \Z$ is $\Z_m$.

The left coset of $k \in \Z$ is denoted $k + m \Z$, which is the same thing as $\eqclass k m$ from the definition of residue class.

So $\index \Z {m \Z} = m$ follows from the definition of Subgroup Index.

$\blacksquare$