Quotient Group of Integers by Multiples
Theorem
Let $\left({\Z, +}\right)$ be the Additive Group of Integers.
Let $\left({m \Z, +}\right)$ be the Additive Group of Integer Multiples of $m$.
Let $\left({\Z_m, +_m}\right)$ be the Additive Group of Integers Modulo $m$.
Then the quotient group of $\left({\Z, +}\right)$ by $\left({m \Z, +}\right)$ is $\left({\Z_m, +_m}\right)$.
Thus $\left[{\Z : m \Z}\right] = m$.
Proof
From Subgroups of the Integers, $\left({m \Z, +}\right)$ is a subgroup of $\left({\Z, +}\right)$.
From All Subgroups of Abelian Group are Normal, $\left({m \Z, +}\right)$ is normal in $\left({\Z, +}\right)$.
Therefore the quotient group $\dfrac {\left({\Z, +}\right)} {\left({m \Z, +}\right)}$ is defined.
Now $\Z$ modulo $m \Z$ is Congruence Modulo a Subgroup.
This is merely congruence of integers as defined in Congruence (Number Theory).
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 $\left[\!\left[{k}\right]\!\right]_m$ from the definition of residue class.
So $\left[{\Z : m \Z}\right] = m$ follows from the definition of Subgroup Index.
$\blacksquare$
Sources
- J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 6.1$: Example $113$
- George McCarty: Topology: An Introduction with Application to Topological Groups (1967): Chapter $\text{II}$
- Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 37$
- Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 39$
- Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 47 \alpha$
- John F. Humphreys: A Course in Group Theory (1996): $\S 7$: Example $7.12$
