\section{Internal Direct Products}

\begin{definition}[Internal Direct Product of $\mathbf{H}$ and $\mathbf{K}$]
	We say that $G$ is the \textit{internal direct product} of $H$ and $K$ and write $G = H \times K$ if $H$ and $K$ are normal subgroups of $G$ and
	\[ G = HK\ \ \ \ \text{and}\ \ \ \ H \cap K = \{e\} \]
\end{definition}

\begin{definition}[Internal Direct Product $\mathbf{H_1 \times H_2 \times \dots \times H_n}$]
	Let $H_1, H_2,\dots,H_n$ be a finite collection of normal subgroups of $G$. We say that $G$ is the \textit{internal direct product} of $H_1,H_2,\dots,H_n$ and write $G=H_1\times H_2 \times \dots \times H_n$, if
	\begin{enumerate}
		\item $G = H_1H_2\dots H_n = \{h_1h_2\dots h_n\ \vert\ h_i \in H_i\}$,
		\item $(H_1H_2\dots H_n) \cap H_{i + 1} = {e}$ for $i=1,2,\dots, n-1$.
	\end{enumerate}
\end{definition}

\begin{theorem}[$\mathbf{H_1 \times H_2 \times \dots \times H_n \approx H_1 \oplus H_2 \oplus \dots \oplus H_n}$]
	If a group $G$ is the internal direct product of a finite number of subgroups $H_1,H_2, \dots, H_n$, then $G$ is isomorphic to the external direct product of $H_1,H_2 \dots, H_n$.
\end{theorem}

\begin{theorem}[Classification of Groups of Order $\mathbf{p^2}$]
	Every group of order $p^2$, where $p$ is a prime, is isomorphic to $\Z_{p^2}$ or $\Z_p \oplus \Z_p$.
\end{theorem}

\begin{corollary}
	If $G$ is a group of order $p^2$, where $p$ is a prime, then $G$ is Abelian.
\end{corollary}
