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\pagestyle{fancy} \lhead{{\sf Homework 6 $|$ Due October 10 (Wednesday)
}} \rhead{\thepage} \cfoot{{\sf Math 8510 $|$ Abstract Algebra I $|$
    Fall 2018 $|$ M.~Macauley}}

\begin{document}

$\;$ \vspace{-4mm}


\begin{enumerate}


\item Let $\mathfrak{C}$ be a category.
  \begin{enumerate}
  \item If $f\in\Hom_{\mathfrak{C}}(A,B)$ is an isomorphism, show that
    there is at most one $g\in\Hom_{\mathfrak{C}}(B,A)$ such that
    $gf=1_A$ and $fg=1_B$.
\item Prove that any two initial (universal) objects in $\mathfrak{C}$
  are equivalent.
\item Prove that any two terminal (couniversal) objects in
  $\mathfrak{C}$ are equivalent. \medskip
\end{enumerate}



\item Let $\{A_i\mid i\in I\}$ be a family of objects in a category
  $\mathfrak{C}$ where products and coproducts always exist. In class,
  we defined a new category $\mathfrak{D}$ where the objects involved
  maps of the form $B\stackrel{f_i}{\longto} A_i$, and morphisms were
  defined so that the product of $\{A_i\mid i\in I\}$ arose as a
  terminal object. Carry out an analogous construction of a category,
  by carefully defining its objects and morphisms, so that the
  coproduct of $\{A_i\mid i\in I\}$ arises as an initial object. Prove
  all of your claims. \medskip



\item Let $A_1$, $A_2$, $A$ be objects in a category $\mathfrak{C}$,
  and $f_i\in\Hom(A,A_i)$ for $i=1,2$. Suppose that \vspace{-2mm}
  \[
  \xymatrix{A\ar@{->}[r]^{f_1}\ar@{->}[d]_{f_2} & A_1\ar@{->}[d]^{g_1} \\
  A_2\ar@{->}[r]_{g_2} & B}\qquad\qquad \text{and}\qquad\qquad 
  \xymatrix{A\ar@{->}[r]^{f_1}\ar@{->}[d]_{f_2} & A_1\ar@{->}[d]^{g'_1} \\
  A_2\ar@{->}[r]_{g'_2} & B'}
  \]
  are pushouts for $(A, A_1,A_2,f_1,f_2)$. Prove that $B$ and $B'$ are
  equivalent. \medskip



\item A pair of homomorphisms $K\stackrel{f}{\to}G\stackrel{g}{\to}H$
  is \emph{exact} at $G$ if $\im(f)=\ker g$. A \emph{short exact
    sequence} is a sequence $1\to
  K\stackrel{f}{\to}G\stackrel{g}{\to}H\to 1$ that is exact at $K$,
  $G$, and $H$.
  \begin{enumerate}
    \item Show that if $K\lhd G$, $f\colon K\to G$ is the inclusion
      map and $g\colon G\to G/K$ is the canonical quotient map, then
      $1\to K\stackrel{f}{\to}G\stackrel{g}{\to}G/K\to 1$ is a short
      exact sequence.
  \item Show that $1\to K\stackrel{f}{\to}G\stackrel{g}{\to}H\to
    1$ is short exact if and only if $f$ is $1$--$1$, $g$ is onto, and
    $\im(f)=\ker g$. Conclude that then $K$ is isomorphic with a normal
    subgroup of $G$ and that $G/f(K)\cong H$.
  \item Suppose $1\to K\to G\to H\to 1$ is a short exact
    sequence. Show that $G$ is solvable if and only if both $K$ and
    $H$ are solvable.
  \item Give an example of a short exact sequence $1\to K\to G\to H\to
    1$ for which $K$ and $H$ are nilpotent but $G$ is not. \medskip
  \end{enumerate}



\item If $G$ is a group and $x\in G$, define the \emph{inner
  automorphism} $f_x$ by setting $f_x(y)=xyx^{-1}$ for all $y\in
  G$. Write $I(G)$ for the set of all inner automorphisms of $G$.
  \begin{enumerate}
    \item Show that $I(G)\leq\Aut(G)$.
    \item Show that $I(G)\cong G/Z(G)$.
    \item If $I(G)$ is abelian show that $G'\leq Z(G)$. Conclude
      that $G$ is nilpotent. 
    \item Determine $\Aut(S_3)$, with proof. [\emph{Hint}: Any
      automorphism must permute the elements of the same order.]
  \end{enumerate}

\end{enumerate}


\end{document}

