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\noindent{\Large{\textsc{Modern Algebra \hfill Assignment 11  \\ Spring 2023 \hfill Due April 5}}}

\large

\vspace{1cm}

\begin{exercise}
Let $p$ be a prime number and let $A$ be an abelian $p$-group.  Show that every subgroup and every quotient of $A$ is also an abelian $p$-group.  [{\em Suggestion.}  If $A' \le A$, the natural map $\pi : A \to A/A'$ is a homomorphism.  Now see Exercise 5.1.4.]
\end{exercise}

\begin{exercise}
Let $p$ be a prime number and define
\[
\Z_{(p)} = \left\{ \frac{n}{p^m} \, \bigg| \, n \in \Z, m \in \N_0  \right\}.
\]

\begin{itemize}
\item[\bf a.]  Show that $\Z_{(p)}$ is a subgroup of $(\Q,+)$ containing $\Z$.

\item[\bf b.]  Show that $Z(p^{\infty}) := \Z_{(p)}/\Z$ is an infinite $p$-group.
\end{itemize}

\end{exercise}

\begin{exercise}
Let $A$ and $B$ be abelian groups, and let $f:A \to B$ be a homomorphism.  Suppose there is a homomorphism $g : B \to A$ so that $f \circ g = 1_B$.  Prove that $A = \ker f \oplus \operatorname{im} g$.  [{\em Suggestion:}  Given $a \in A$, write $a = (a - g(f(a))) + g(f(a))$.]
\end{exercise}


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