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\begin{document}
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\begin{Large}
{\bf Math 3140 - Assignment 10}
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Due November 2, 2016
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\begin{enumerate}

\item
 Find all abelian groups of order $180$ up to isomorphism.

\item
 How many abelian groups are there of order $3^5$ up to isomorphism?
 How many of order $p^5$ where $p$ is prime?

\item
 Let $(A,+)$ be an abelian group, $a_1,\dots,a_n\in A$. Show that
\[ B = \{z_1 a_1+\dots+z_n a_n \setsuchthat z_1,\dots,z_n\in\Z \} \]
 is a subgroup of $(A,+)$.


\item
 Let $(A,+)$ be an abelian group. The set 
\[ A_{\mathrm{tor}} := \{ x\in A \setsuchthat o(x) \text{ is finite}\} \]
 of elements of finite order is called the \emph{torsion part} of $A$.

 Show that $A_{\mathrm{tor}}$ is a subgroup of $(A,+)$.

\item
 Let $A = \Z_{n_1}\times\dots\times\Z_{n_k}\times\Z^r$.
\begin{enumerate}
\item What is the torsion part of $A$?
\item What is $A/A_{\mathrm{tor}}$ isomorphic to?
\end{enumerate}




%\item
% Show that for $m,n\in\N$ relatively prime the multiplicative groups $\Z_{mn}^*$ and $\Z_m^*\times\Z_n^*$
% are isomorphic.

\end{enumerate}




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