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\begin{document}
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{\Large \textmd{MA434: Spring 2019}}

\noindent
{\large \textmd{Homework 7}}

\noindent
{\bf Contact:} \texttt{gwmelvin\makeatletter@\makeatother colby.edu}


\end{flushright}
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\noindent
\textbf{Some thoughts and advice:} 
\begin{itemize}
\item Please submit solutions to the following problems by {\bf Wednesday, April 3rd, 1.10pm}. You can either submit your solution in class or leave it outside my office.
\item You should expect to spend several hours on homework sets. A lot of practice problem-solving is essential to understand the material and skills covered in class. Be organised and do not leave problem sets until the last-minute. Instead, get a good start on the problems as soon as possible. 
\item When approaching a problem think about the following: \textsl{do you understand the words used to state the problem? what is the problem asking you to do? can you restate the problem in your own words? have you seen a similar problem worked out in class? is there a similar problem worked out in the textbook? what results/skills did you see in class that might be related to the problem?}

If you are stuck for inspiration come to office hours, or send me an email. However, don't just ask for the solution - provide your thought process, the difficulties you are having, and ask a coherent question in complete English sentences. 
\item Form study groups - get together and work through problem sets. \textbf{This will make your life easier!} You must write your solutions \textsl{on your own} and \textsl{in your own words}.
\item You {\bf are not allowed} to use any additional resources (e.g. \texttt{stackoverflow.com}).  If you are concerned then please ask.
\end{itemize}
\hrule

\medskip
{\large {\bf Some Group Theory \& Schur's Lemma} }

\medskip
{\sc Unless otherwise specified, all vector spaces will have scalar field $\C$.}

\medskip
\noindent
{\bf Some Definitions/Theorems:}
\begin{itemize}
\item (First Isomorphism Theorem) {\sl Let $f:G\rightarrow H$ be a group homomorphism. Then, $G/\ker f \simeq \im f$.}
\item (Fundamental Theorem of Finite Abelian Groups) {\sl Let $G$ be a finite abelian group. Then, there exists integers $n_{1},\ldots, n_{r}$ such that }
	\[ G\simeq \Z/n_{1}\Z\times \Z/n_{2}\Z\times \cdots \times \Z/n_{r}\Z\]
	{\sl where $n_{i}$ divides $n_{i+1}$, for all $i=1,\ldots,r-1$.}
\item A representation $\rho:G\rightarrow \GL(V)$ is said to be {\bf faithful} if $\ker \rho=\{e_{G}\}$ i.e. $\rho$ is injective.
\item The {\bf centre of $G$} is $Z(G) = \{z\in G\;|\; zg=gz \text{ for all $g\in G$}\}$.	
\end{itemize}

\noindent
{\bf Problems:}
\begin{enumerate}
\item For every $g\in G$, define $c_{g}:G\rightarrow G\;,\; h\mapsto ghg^{-1}$.
\begin{enumerate}
\item Prove directly that $c_{g}$ is an automorphism (i.e. a bijective homomorphism) of $G$, for every $g\in G$. 
\item What is the inverse $c_{g}^{-1}$?	
\item Denote the group of all automorphisms of $G$ by $\Aut(G)$. Show that 
	\[C:G\rightarrow \Aut(G)\;,\; g\mapsto c_{g}\]
is a homomorphism.
\item Using the previous exercise, show that $Z(G)$ is a normal subgroup of $G$. ({\sl Hint: what is $\ker C$?})
\end{enumerate}
\item Let $G$ be a finite subgroup of $\C^{\times}=\{z\in \C\;| \;z\neq 0\}$. Here the group operation on $\C^{\times}$ is multiplication. In particular, $G$ is a finite abelian group. Denote $n=|G|$, the order of $G$.
\begin{enumerate}
\item Let $p(t)=t^{n}-1\in \C[t]$. Show that $p(g)=0$, for every $g\in G$.
\item Let $k=\max\{o(g)\;|\; g\in G\}$, where $o(g)=|g|$ is the order of $g\in G$. Prove that $k=n$. ({\sl Hint: proceed by contradiction and use the Fundamental Theorem of Finite Abelian Groups}) 
\item Deduce the following: {\sl any finite subgroup of $\C^{\times}$ is cyclic.} ({\sl Hint: if $f\in \C[t]$ and $\deg f = r$ then $f$ has at most $r$ roots.})
\end{enumerate}

{\bf Remark:} There is nothing special about $\C$ here: the same argument shows that, for $k$ any field, any finite subgroup of $K^{\times}$ is cyclic. In particular, if $k=\Z/p\Z$ then $k^{\times}$ is cyclic: this has important consequences for cryptography.

	


\item  Suppose that $(\rho,V)$ is a faithful representation of $G$.
\begin{enumerate}
\item Let $z\in Z(G)$. Show that $\rho_{z}\in \End_{G}(\rho)$.
\item Using the First Isomorphism Theorem, prove that $Z(G)$ is isomorphic to a finite subgroup of $\GL_{G}(\rho)=\{T\in \End_{G}(\rho)\;|\; T\text{ invertible}\}$.
\item Deduce the following: {\sl if $G$ admits a faithful, irreducible representation then $Z(G)$ is cyclic.}
\end{enumerate}
\end{enumerate}












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