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\newtheorem{prob}{Problem}
\newtheorem{exc}[prob]{Exercise}
\newtheorem{chal}[prob]{Challenge}
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\title{Math 332: Homework 6}
%\author{Your name here --- delete the % at the start of the line}
\date{16.III.15}

\begin{document}
\maketitle

\begin{exc}
Construct your own \emph{handcrafted, artisanal} semi-direct product:  carefully select groups $H$ and $K$ from a small batch producer in the Cascades.  Choose an explicit nontrivial homomorphism $\varphi:K\to \Aut(H)$.  Write the product on $H\semi K$ explicitly.  If you can, find an isomorphism from $H\semi K$ to a more familiar group.  (Since your $H\semi K$ is \emph{handcrafted} and \emph{artisanal}, do not choose an example from the book or class.)
\end{exc}

\begin{prob}\label{upper}
Let $F$ be a field and let $G$ denote the group of upper triangular matrices in $GL_n(F)$.  Let $U$ denote matrices in $G$ with $1$'s along the diagonal, and let $D$ denote diagonal matrices in $G$ (arbitrary nonzero elements of $F$ along the diagonal, $0$'s elsewhere).  Prove that
\[
  G\cong U\semi D.
\]
\end{prob}

\begin{prob}
In the setup of Problem \ref{upper}, let $n=2$.  In this case, $D\cong F^\times\times F^\times$ and $U\cong F$ (where $F$ refers to the additive group of $F$).  Explicitly determine the homomorphism
\[
  \varphi:F^\times\times F^\times\to \Aut(F)
\]
which expresses $G$ as the semi-direct product $U\semi D$.
\end{prob}

\begin{prob}
Prove that $SL_2(\FF_3)$ is isomorphic to the semi-direct product of $Z_3$ and $Q_8$.  (It's up to you to determine if it's $Z_3\semi Q_8$ or $Q_8\semi Z_3$!)
\end{prob}

\begin{prob}
Let $F_1$ and $F_2$ be free groups of the same finite free rank.  Prove that $F_1\cong F_2$.  (\emph{Bonus}: Do the same when $F_1$ and $F_2$ have free ranks of the same infinite cardinality.)
\end{prob}

\begin{prob}
Find a presentation of $A_4$ with two generators (and prove that it is indeed a presentation of $A_4$).
\end{prob}

\begin{chal}
Define the \emph{free abelian group} on the set $S$ to be
\[
  \ZZ(S) = \langle S \mid [a,b]=1\text{ for all }a,b\in S\rangle.
\]
Invent the universal property of $\ZZ(S)$ and prove that $\ZZ(S)$ satisfies it.  Use the universal property to prove that $\ZZ(\{1,2,\ldots,n\})$ is the $n$-fold direct product of $\ZZ$ with itself.
\end{chal}

\end{document}