Integers Modulo m under Addition form Cyclic Group

Theorem
Let $\Z_m$ be the set of integers modulo $m$.

Let $+_m$ be the operation of addition modulo $m$.

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

Then $\struct {\Z_m, +_m}$ is a cyclic group of order $m$, generated by the element $\eqclass 1 m \in \Z_m$.

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


 * $\Z_m = \dfrac \Z {\mathcal R_m} = \set {\eqclass 0 m, \eqclass 1 m, \ldots, \eqclass {m - 1} m}$

It is established that modulo addition is well defined:


 * $\eqclass a m +_m \eqclass b m = \eqclass {a + b} m$

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 $\struct {\Z_m, +_m}$ is $\eqclass 0 m$.
 * G3: Inverses: The inverse of $\eqclass k m \in \Z_m$ is $-\eqclass k m = \eqclass {-k} m = \eqclass {n - k} m$.
 * Commutativity: Addition modulo $m$ is commutative.

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

Also see

 * Integers Modulo m under Addition form Abelian Group