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\noindent{\Large{\textsc{Modern Algebra \hfill Assignment 5.2  \\ Spring 2023 \hfill Due February 15}}}

\large

\vspace{1cm}

\begin{exercise}
Define $f : D_n \to \{ \pm 1 \}$ by
\[
f(x) = \begin{cases} 1 & \text{ if } x \text{ is a rotation,} \\ -1 & \text{ otherwise.} \end{cases}
\]
Prove that $f$ is a homomorphism.  What is its kernel?
\end{exercise}

\begin{exercise}
Lang, Exercise II.3.11.
\end{exercise}

\begin{exercise}
Let $A$ be an abelian group and let $n \in \N$.  We say that $n$ is an {\em exponent} for $A$ provided $na = 0$ for all $a \in A$.\footnote{The terminology is borrowed from the equivalent multiplicative formulation $a^n = e$ and is entirely standard, if not a little confusing.}  Given $m \in \N$, recall that the $m$-torsion subgroup of $A$ is
\[
A[m] = \{ a \in A \, | ma = 0 \}.
\]  
So $n$ is an exponent for $A$ if and only if $A = A[n]$.  Assume that this is the case in {\bf a}--{\bf c} below.
\begin{itemize}

\item[\bf a.]  Suppose $n=rs$ for some $r,s \in \N$.  Let $E_r : A \to A$ be the homomorphism defined by $E_r(a) = ra$.  Show that $\operatorname{im} E_r <A \left[ s \right]$.

\item[\bf b.]  Assuming that we also have $\gcd(r,s)=1$, prove that $\operatorname{im} E_r = A \left[ s \right]$.  [{\em Suggestion.}  Let $a \in A[s]$ and use B\'ezout's lemma to write $ur + vs = 1$ with $u,v \in \Z$.  Then consider the equation $a = 1a$.]

\item[\bf c.]  Prove that the map $f :A \to A[r] \oplus A[s]$ given by $f = E_s \oplus E_r$ is an isomorphism.  [{\em Suggestion.}  Use the previous suggestion to show $\ker f$ is trivial.  Given $(b,c) \in A[r] \oplus A[s]$, show that $f(uc + vb) = (b,c)$.]

\end{itemize}


\end{exercise}

\begin{exercise}
Let $A$ be an (additive) abelian group and $B, C < A$.  Define $f: B \times C \to A$ by $f(b,c) = b+c$.

\begin{itemize}

\item[\bf a.]  Prove that $f$ is a homomorphism.

\item[\bf b.]  Use part {\bf{a}} to prove that $B + C = \{b + c \, | \, b \in B, c \in C \}$ is a subgroup of $A$.

\item[\bf c.]  Compute $\operatorname{ker} f$.

\end{itemize}

\end{exercise}


%\begin{exercise}
%Let $A$ be an abelian group, written additively.

%\begin{itemize}

%\item[\bf a.]  Let $a,b \in A$ and $n \in \Z$.  Prove that $n(a+b) = na + nb$.  [{\em Suggestion:}  Treat the cases $n>0$, $n=0$ and $n<0$ separately.  Induction may be helpful.]

%\item[\bf b.]  For $n \in \N$, let  $A[n] = \{ a \in A \, | \, na = 0 \}$.  Prove that $A[n] < A$.  $A[n]$ is called the {\em $n$-torsion subgroup} of $A$.

%\item[\bf c.] Let
%\[
%A_{\text{tors}} = \bigcup_{n \in \N} A[n].
%\]
%Prove that $A_{\text{tors}} < A$.  $A_{\text{tors}}$ is called the {\em torsion subgroup} of $A$.

%\end{itemize}

%\end{exercise}

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