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:


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

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 Infinite Cyclic Group and Quotient Group of Cyclic Group, $$\left({\frac {\Z} {\mathcal R_m}, +_m}\right)$$ is cyclic order $$m$$.