Additive Group of Integers Modulo m
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:
- $\displaystyle \Z_m = \frac {\Z} {\mathcal R_m} = \left\{{\left[\!\left[{0}\right]\!\right]_m, \left[\!\left[{1}\right]\!\right]_m, \ldots, \left[\!\left[{m-1}\right]\!\right]_m}\right\}$
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, $\displaystyle \left({\frac {\Z} {\mathcal R_m}, +_m}\right)$ is cyclic order $m$.
$\blacksquare$
Needs examination to see whether it can be extended to apply to all $m \in \R$.
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Sources
- Seth Warner: Modern Algebra (1965): $\S 7$: Example $7.1$
- Seth Warner: Modern Algebra (1965): $\S 25$: Theorem $25.4$
- George McCarty: Topology: An Introduction with Application to Topological Groups (1967): Chapter $\text{II}$
- Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 33$
- Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 43$
- Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 34 \ (3)$
- John F. Humphreys: A Course in Group Theory (1996): $\S 2$: Corollary $2.32$
