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\noindent{\Large{\textsc{Modern Algebra \hfill Assignment 6.2  \\ Spring 2023 \hfill Due February 22}}}

\large

\vspace{1cm}

\begin{exercise}
Let $n \in \N$.  Compute $[\Z:n\Z]$.  [{\em Suggestion.}  Remember that the cosets of $n\Z$ in $\Z$ are precisely the congruence classes modulo $n$].
\end{exercise}

\begin{exercise}
Compute $[\R^{\times} : \R^+]$.
\end{exercise}

\begin{exercise}
Show that every coset of $\R^+$ in $\C^{\times}$ contains a unique element of absolute value equal to 1.  Conversely, show that every coset of $S^1 = \{ z \in \C \, | \, |z| = 1\}$ in $\C^{\times}$ contains a unique positive real number.  In both cases, describe the cosets geometrically.
\end{exercise}

\begin{exercise}
Prove that $[\R:\Q]$ is infinite.  [{\em Suggestion:}  Argue by contradiction.]
\end{exercise}

\begin{exercise}
Let $G$ be a group of order $pqr$, where $p$, $q$ and $r$ are distinct primes.  If $H, K < G$ satisfy $|H|=qp$ and $|K|=qr$, prove that $|H \cap K| = q$. [{\em Suggestion:}  Observe that $K$ has more elements than $H$ has (left) cosets, then use the pigeonhole principle.]
\end{exercise}

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