Subgroup of Additive Group of Integers Generated by Two Integers
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Theorem
Let $m, n \in \Z_{> 0}$ be (strictly) positive integers.
Let $\struct {\Z, +}$ denote the additive group of integers.
Let $\gen {m, n}$ be the subgroup of $\struct {\Z, +}$ generated by $m$ and $n$.
Then:
- $\gen {m, n} = \struct {\gcd \set {m, n} \Z, +}$
That is, the additive groups of integer multiples of $\gcd \set {m, n}$, where $\gcd \set {m, n}$ is the greatest common divisor of $m$ and $n$.
Proof
By definition:
- $\gen {m, n} = \set {x \in \Z: \gcd \set {m, n} \divides x}$
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Hence the result.
$\blacksquare$
Sources
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $5$: Subgroups: Exercise $10$
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Subgroups and Cosets: $\S 36 \beta$