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\noindent{\Large{\textsc{Modern Algebra \hfill Assignment 13.2  \\ Spring 2023 \hfill Due April 19}}}

\large

\vspace{1cm}


\begin{exercise}
Let $A$ be an additive abelian group and for $k \in \N$ let $C_k$ denote a cyclic subgroup of $A$ with order $k$.

\begin{itemize}

\item[\bf a.]  Let $m,n \in \N$ and suppose $\gcd(m,n) = 1$.  Use Lagrange's Theorem to show that the sum $C_m + C_n$ is direct.

\item[\bf b.]  With $m$ and $n$ as above, write $C_m = \langle a \rangle$ and $C_n = \langle b \rangle$.  Show that $a+b$ has order $mn$.  Conclude that $C_m \oplus C_n = C_{mn}$.

\item[\bf c.]  Let $m_1, m_2, \ldots, m_k \in \N$ and suppose $\gcd(m_i, m_j) = 1$ for all $i \ne j$.  Use induction to prove that
\[
C_{m_1} \oplus \cdots \oplus C_{m_k} = C_{m_1 m_2 \cdots m_k}.
\]


\end{itemize}

\end{exercise}


\begin{exercise}
Classify the finite abelian groups of order 4312 (there are 6, up to isomorphism).
\end{exercise}


\begin{exercise}
Lang, exercise II.7.2.
\end{exercise}


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