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\begin{center}
Math 601 Midterm\\
October 9, 2000\\
%Karen A. Chandler \\
\end{center}

{\bf Instructions:}
Please do all the questions. 
Please do not hesitate to ask for clarification.
You may use the textbook and your notes for reference.

Good luck!


\begin{enumerate}

\item Suppose that $G$ is a finite group and $H$ is a normal subgroup
of $G$ so that $[G:H]$ is odd.  Suppose further $G$ has a 
subgroup~$K$ of order~8.  Show that~$K \subseteq H$.

\item Let~$G$ be a group.
Let~$S=\{ghg^{-1} h^{-1}: g, h \in G \}$.
Let~$C=<S>$.

($C$ is called the {\bf commutator} subgroup of G.)
\begin{enumerate}
\item Show that $C=\{s_1 s_2 \ldots s_n : s_1, \ldots, s_n \in S\}$.
\item Show that $C$ is a  normal subgroup of~$G$.

(Suggestion: start by considering each element of $S$, then use (a).)
\item Show that~$G/C$ is abelian.
\item Suppose that~$\Phi: G \rightarrow A$ is a homomorphism
between~$G$ and an abelian group~$A$.
Let $\pi: G \rightarrow G/C$ be the canonical projection.
Show that there is a unique homomorphism $\tilde \Phi: G/C \rightarrow A$
so that $\Phi = \tilde \Phi \circ  \pi$:
\end{enumerate}


\item Let $n \in \nats$. 
Let $G$ be, as a  set (but {\em not} as a group), the Cartesian product
$\ints/2 \ints \times \ints/n \ints$.  

Define a binary operation on $G$ by 
$$(\bar i, \bar j) (\bar k, \bar m)=
 (\bar i + \bar k, (-1)^k \bar j+\bar m).$$ 
\begin{enumerate}
\item Show that $G$ is a group.
\item Verify that $G$ is an internal semidirect product of
its cyclic subgroups 
$\ints/2 \ints \times \{\bar 0\}$
\linebreak
 and~${\{\bar 0\} \times \ints/n \ints}$.

(Recall that $G$ is an internal semidirect product of two subgroups~$H$
and $N$
if $N$ is normal, $G=HN$ and $H \cap N=\{e\}$.)
\item Do you recognise this group?
\end{enumerate}

\item Let $H$ be a group.
\begin{enumerate}
\item Show that there is a category ${\cal C}$ in which:
\begin{itemize}
\item  an object is
a group containing $H$ as a subgroup, and
\item a morphism is a group homomorphism that restricts to the identity
on~$H$.
\end{itemize}
\item Show that for each object $ G$ of  ${\cal C}$ there is a unique
morphism  $H \rightarrow G$.

(So $H$ is an {\bf initial} (or {\bf universal} or {\bf universally repelling})
object in the category.)
\item How would (a) and/or (b) change if we sought a category in which
an object is a group having a subgroup isomorphic to~$H$?
\end{enumerate}

\end{enumerate}




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