Give examples of groups with the following properties. Feel free to make them as simple as possible.

1. A finite abelian group
2. A finite nonabelian group
3. A finite group with exactly one proper nontrivial subgroup
4. An infinite abelian group
5. An infinite nonabelian group
6. An infinite group in which no nontrivial element has finite order
7. An infinite group in which every element has finite order
8. An infinite group with exactly one nontrivial element of finite order

• None.

problem.examples_of_abstract_groups.tex

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\begin{problem}
Give examples of groups with the following properties. Feel free to make them as simple as possible.
\begin{enumerate}
\item A finite abelian group
\item A finite nonabelian group
\item A finite group with exactly one  proper nontrivial subgroup
\item An infinite abelian group
\item An infinite nonabelian group
\item An infinite group in which no nontrivial element has finite order
\item An infinite group in which every element has finite order
\item An infinite group with exactly one nontrivial element of finite order
\end{enumerate}
\end{problem}

• None.