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\begin{problem}
Give examples of groups with the following properties by \textbf{explicitly} giving a set and defining both functions as well as the distinguished element. 
\begin{enumerate}
\item A group with $3$ elements
\item A different group with $3$ elements, if possible
\item A group with $4$ elements
\item A different group with $4$ elements, if possible
\item An infinite group
\end{enumerate}
\end{problem}