Subgroups of Additive Group of Integers Modulo m
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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: $\S 36 \gamma$