Order of Additive Group of Integers Modulo m

Theorem

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

The order of $\struct {\Z_m, +_m}$ is $m$.

Proof

By definition, the order of a group is the cardinality of its underlying set.

By definition, the underlying set of $\struct {\Z_m, +_m}$ is the set of residue classes $\Z_m$:

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

From Cardinality of Set of Residue Classes, $\Z_m$ has $m$ elements.

Hence the result.

$\blacksquare$

Sources

• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Subgroups and Cosets: $\S 38$