\section{Properties of External Direct Products}

\begin{theorem}[Order of an Element in a Direct Product]
	The order of an element in a direct product of a finite number of finite groups is the least common multiple of the orders of the component of the element. In symbols,
	\[ \abs{(g_1,g_2,\dots,g_n)} = \lcm(\abs{g_1},\abs{g_2},\dots,\abs{g_n}) \]
\end{theorem}

\begin{theorem}[Criterion for $\mathbf{G \oplus H}$ to be Cyclic]
	Let $G$ and $H$ be finite cyclic groups. Then $G \oplus H$ is cyclic if and only if $\abs{G}$ and $\abs{H}$ are relatively prime.
\end{theorem}

\begin{corollary}[Criterion for $\mathbf{G_1 \oplus G_2 \oplus \dots \oplus G_n}$ to Be Cyclic]
	An external direct product $G_1 \oplus G_2 \oplus \dots \oplus G_n$ of a finite number of finite cyclic groups is cyclic if and only if $\abs{G_i}$ and $\abs{G_j}$ are relatively prime when $i \neq j$.
\end{corollary}

\begin{corollary}[Criterion for $\mathbf{\Z_{n_1n_2\dots n_k} \approx \Z_{n_1} \oplus \Z_{n_2} \oplus \dots \oplus \Z_{n_k}}$]
	Let $m = n_1n_2\dots n_k$. Then $\Z_m$ is isomorphic to $\Z_{n_1} \oplus \Z_{n_2} \oplus \dots \oplus \Z_{n_k}$ if and only if $n_i$ and $n_j$ are relatively prime when $i \neq j$.
\end{corollary}
