\section{Normal Subgroups}

\begin{definition}[Normal Subgroup]
	A subgroup $H$ of a group $G$ is called a \textit{normal} subgroup of $G$ if $aH = Ha$ for all $a$ in $G$. We denote this by $H \triangleleft G$.
\end{definition}

\begin{theorem}[Normal Subgroup Test]
	A subgroup $H$ of $G$ is normal in $G$ if and only if $xHx^{-1} \subseteq H$ for all $x$ in $G$.
\end{theorem}