Subgroups of Additive Group of Integers Modulo m

Theorem

Let $n \in \Z_{> 0}$ be a (strictly) positive integer.

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

The subgroups of $\struct {\Z_m, +_m}$ are the additive groups of integers modulo $k$ where:

$k \divides m$

Proof

From Integers Modulo m under Addition form Cyclic Group, $\struct {\Z_m, +_m}$ is cyclic.

Let $H$ be a subgroup of $\struct {\Z_m, +_m}$

From Subgroup of Cyclic Group is Cyclic, $H$ is of the form $\struct {\Z_k, +_k}$ for some $k \in \Z$.

From Lagrange's Theorem, it follows that $k \divides m$.

Hence the result.

$\blacksquare$

Sources

• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Subgroups and Cosets: $\S 36 \gamma$