Integers Modulo m under Addition form Cyclic Group

Theorem
The structure $\left({\Z_m, +_m}\right)$ where $\Z_m$ is the set of integers modulo $m$, is an abelian group.

$\left({\Z_m, +_m}\right)$ is cyclic of order $m$, generated by the element $\left[\!\left[{1}\right]\!\right]_m \in \Z_m$.

Frequently, the operation $+_m$ (addition modulo $m$) is merely written $+$, as long as it is understood what this operation actually is.

Proof
From the definition of integers modulo $m$, we have:


 * $\Z_m = \dfrac \Z {\mathcal R_m} = \left\{{\left[\!\left[{0}\right]\!\right]_m, \left[\!\left[{1}\right]\!\right]_m, \ldots, \left[\!\left[{m-1}\right]\!\right]_m}\right\}$

It is established that modulo addition is well defined:


 * $\left[\!\left[{a}\right]\!\right]_m +_m \left[\!\left[{b}\right]\!\right]_m = \left[\!\left[{a + b}\right]\!\right]_m$

The group axioms are fulfilled:


 * G0: Closure: Addition modulo $m$ is closed.
 * G1: Associativity: Addition modulo $m$ is associative.
 * G2: Identity: The identity element of $\left({\Z_m, +_m}\right)$ is $\left[\!\left[{0}\right]\!\right]_m$.
 * G3: Inverses: The inverse of $\left[\!\left[{k}\right]\!\right]_m \in \Z_m$ is $- \left[\!\left[{k}\right]\!\right]_m = \left[\!\left[{-k}\right]\!\right]_m = \left[\!\left[{n - k}\right]\!\right]_m$.
 * Commutativity: Addition modulo $m$ is commutative.

From Integers under Addition form Infinite Cyclic Group and Quotient Group of Cyclic Group, $\left({\dfrac \Z {\mathcal R_m}, +_m}\right)$ is cyclic order $m$.