Intersection of Multiplicative Groups of Complex Roots of Unity

Theorem

Let $\struct {K, \times}$ denote the circle group.

Let $m, n \in \Z_{>0}$ be (strictly) positive integers.

Let $c = \lcm \set {m, n}$ be the lowest common multiple of $m$ and $n$.

Let $\struct {U_n, \times}$ denote the multiplicative group of complex $n$th roots of unity.

Let $\struct {U_m, \times}$ denote the multiplicative group of complex $m$th roots of unity.

Let $H = U_m \cap U_n$.

Then $H = U_c$.

Proof

This theorem requires a proof. In particular: This is (probably) a specialisation of a more general result on cyclic groups. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Subgroups and Cosets: $\S 44 \gamma$