Integers Modulo m under Addition form Cyclic Group

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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:

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

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:

Group Axiom $\text G 0$: Closure: Addition modulo $m$ is closed.
Group Axiom $\text G 1$: Associativity: Addition modulo $m$ is associative.
Group Axiom $\text G 2$: Existence of Identity Element: The identity element of $\struct {\Z_m, +_m}$ is $\eqclass 0 m$.
Group Axiom $\text G 3$: Existence of Inverse Element: 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, $\struct {\dfrac \Z {\RR_m}, +_m}$ is cyclic order $m$.

$\blacksquare$


Also see


Sources

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