\part{Product Groups}

\chapter{Direct Product Groups}

\begin{definition}
A \emph{product of group subsets} is the set closed by multiplying elements of two subsets together.
\[ S,T \subseteq G \implies ST= \{ s \cdot t : s \in S \land t \in T \} \]
\end{definition}

\begin{definition}
	A \emph{product group} is a group $(G \times H, \cdot)$ formed by the groups $(G,\cdot_G)$ and $(H,\cdot_H)$ with the operation as such.
\[ (g_1,h_1) \cdot (g_2,h_2)=(g_1\cdot_G g_2 , h_1 \cdot_H h_2) \]
\begin{itemize}
	\item $G \times H$ is the cartesian product of the group sets
	\item $G \times H$ is the cartesian product of the group sets
\end{itemize}
\end{definition}


\begin{proposition}
\[ M,N \lhd G \land M \cap N = \{1_G\} \implies NM \leq G \land M\times N \cong MN \]
\end{proposition}

\begin{proposition}
	\[ H,K \leq G \land H\times K \cong G \iff H \cap K =\{1_G\} \land HK=G \land K \lhd G \]
\end{proposition}

\section{A 'Chinese remainder theorem' for groups}
\begin{proposition}Let $(n_i)_{i=1}^{r} \in \mathbb{Z}$ be coprime integers and $N= \prod^{r}_{i=1} n_i$. Then $ \mathbb{Z} \\ N \mathbb{Z} \cong \large\times^{r}_{i=1} \mathbb{Z} / n_i \mathbb{Z}$.
\end{proposition}
\chapter{Semidirect Product Groups}
- semidirect product group
\section{Relationship between Direct and Semidirect Product Groups}
- when is a direct-semidirect map an isomorphism? 
