Integers Modulo m under Addition form Cyclic Group

From ProofWiki
Jump to navigation Jump to search

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 $m$.

$\blacksquare$


Also see


Sources