Some explicit examples of abstract groups

Give examples of groups with the following properties by giving a set and defining both functions as well as the distinguished element.

1. A group with elements
2. A different group with elements, if possible
3. A group with elements
4. A different group with elements, if possible
5. An infinite group

• None.

problem.explicit_examples_of_abstract_groups.tex

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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}

Did you decide to use your definition that used to be in the wiki? I would potentially change this to be

• by giving a set, defining the binary operation, stating the inverse of each element, and identifying the identity.
• Do you want them to show that the group axioms hold? I could see a student doing all the above but failing to check associativity. They'll get lucky most likely with small sets…