Group Direct Product of Cyclic Groups

Theorem
Let $G$ and $H$ both be finite cyclic groups whose orders are coprime, i.e. $\left\vert{G}\right\vert \perp \left\vert{H}\right\vert$.

Then their group direct product $G \times H$ is cyclic.

Proof
Let $G$ and $H$ be groups whose identities are $e_G$ and $e_H$ respectively.

Suppose:


 * $(1): \quad \left\vert{G}\right\vert = n, G = \left \langle {x} \right \rangle$
 * $(2): \quad \left\vert{H}\right\vert = m, H = \left \langle {y} \right \rangle$
 * $(3): \quad m \perp n$

Then:

But then $\left({x, y}\right)^{n m} = e_{G \times H} = \left({x^{n m}, y^{n m}}\right)$ and thus $k \mathop \backslash n m$.

So $\left\vert{\left({x, y}\right)}\right\vert = n m \implies \left \langle{\left({x, y}\right)}\right \rangle = G \times H$.