Intersection of Additive Groups of Integer Multiples
Jump to navigation
Jump to search
Theorem
Let $m, n \in \Z_{> 0}$ be (strictly) positive integers.
Let $\struct {m \Z, +}$ and $\struct {n \Z, +}$ be the corresponding additive groups of integer multiples.
Then:
- $\struct {m \Z, +} \cap \struct {n \Z, +} = \struct {\lcm \set {m, n} \Z, +}$
Proof
By definition:
- $m \Z = \set {x \in \Z: m \divides x}$
Thus:
\(\ds m \Z \cap n \Z\) | \(=\) | \(\ds \set {x \in \Z: n \divides x} \cap \set {x \in \Z: m \divides x}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {x \in \Z: n \divides x \land m \divides x}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {x \in \Z: \lcm \set {m, n} \divides x}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \lcm \set {m, n} \Z\) |
Hence the result.
$\blacksquare$
Sources
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $5$: Subgroups: Exercise $5$
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Subgroups and Cosets: $\S 36 \alpha$