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\noindent{\Large{\textsc{Modern Algebra \hfill Assignment 3.2  \\ Spring 2023 \hfill Due February 1}}}

\large

\vspace{1cm}

\begin{exercise}
Let $G$ and $H$ be groups.  If $g \in G$ and $h \in H$ both have finite order, show that in $G \times H$ one has
\[
|(g,h)| = \operatorname{lcm}( |g|, |h|).
\]
\end{exercise}

\begin{exercise}
Lang, exercise II.1.13.
\end{exercise}

\begin{exercise}
If $G$ is a group and $\{H_i \,| \, i \in I \}$ is a collection of subgroups of $G$, prove that
\[
\bigcap_{i \in I} H_i
\]
is a subgroup of $G$.  That is, an (arbitrary) intersection of subgroups of $G$ is again a subgroup of $G$.
\end{exercise}

\begin{exercise}
Show that the subset $R$ of rotations in $D_n$ is a subgroup of $D_n$.  Which subgroups of $D_n$ contain only the identity and flips?
\end{exercise}

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