\section{Definition and Examples}

\begin{definition}[Group Homomorphism]
	A \textit{homomorphism} $\phi$ from a group $G$ to a group $\overline{G}$ is a mapping from $G$ into $\overline{G}$ that preserves the group operation; that is, $\phi(ab) = \phi(a)\phi(b)$ for all $a, b$ in $G$.
\end{definition}

\begin{definition}[Kernel of a Homomorphism]
	The \textit{kernel} of a homomorphism $\phi$ from a group $G$ to a group with identity $e$ is the set $\{x \in G\ \vert\ \phi(x)=e\}$. The kernel of $\phi$ is denoted by $\ker\phi$.
\end{definition}