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\hfill \textbf{March  23, 2015}

\vspace{0.5 cm} \centerline{ \textbf{\large{Homework 3\hspace{1 cm}
Math 6303}}}

\vspace {1 cm}
\begin{enumerate}

\item Prove that a finite abelian group that is not cyclic contains a subgroup which is isomorphic to $\mbb{Z}_p\oplus \mbb{Z}_p$
\item Let $A$ be an abelian group of order $n$ and let $m$ be an integer that divides $n$.  Prove that $A$ contains a subgroup of order $m$.
\item
A finite  abelian group is directly irreducible if it is not the direct sum of abelian groups of smaller order. Give a direct proof  that every finite abelian group is the direct sum of directly irreducible abelian groups. Characterize the directly irreducible abelian groups.
\item Let $A,B,C$ be finite abelian groups. Show that if $A\oplus B\cong A\oplus C$ then $B\cong C$.
\item Let $n$ be a positive integer. Show that every abelian group of order $n$ is cyclic iff $n$ is not divisible by the square of any prime.


\end{enumerate}

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$(n,m)=(n,n)+(m-n)(0,1)$
List all subgroups of order $p^2$ of the abelian group $\mbb{Z}_{p^3}\oplus \mbb{Z}_{p^2}$