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\begin{document}

\textbf{Math 6310, Assignment 4 \hfill Due in class: Monday, October 26}

\medskip 

{\large Name: }

\begin{enumerate}
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\item Let $x,y$ be group elements such that $x$ and $y$ commute with $[x,y]=xyx^{-1}y^{-1}$. Prove that for each $n\ge1$, one has
\[
[x^n,y]=[x,y]^n\qquad\text{ and }\qquad x^ny^n=(xy)^n[x,y]^{\binom{n}{2}}\,.
\]

%\textbf{Solution:} 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item
Let $\FF$ be a field with more than three elements. Prove that the commutator subgroup of $\SL_2(\FF)$ contains all matrices of the form $\displaystyle{\begin{pmatrix} 1&a\cr 0&1\end{pmatrix}}$, for $a\in\FF$.

%\textbf{Solution:} 

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\item For $p,q$ distinct primes, prove that a group of order $p^2q$ is solvable.

%\textbf{Solution:} 

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\item Let $p$ be a prime integer. What is the number of Sylow $p$-subgroups of $\GL_2(\FF_p)$\,?

%\textbf{Solution:} 

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\item For subgroups $H,K$ of $G$ we use $[H,K]$ for the subgroup generated by $hkh^{-1}k^{-1}$ with $h\in H$ and $k\in K$. Prove:
\begin{enumerate}
\item $[H,K] \nor \la H,K\ra$, where $\la H,K\ra$ is the subgroup generated by $H$ and $K$.
\item $[G,H]H \nor G$.
\item $[G,H]H$ is the smallest normal subgroup of $G$ that contains $H$.
\end{enumerate}

%\textbf{Solution:} 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item Let $G$ be a group. Set $Z_0=\{ e\}$ and recall that $Z_{i+1}$ is the inverse image of $Z(G/Z_i)$ under the canonical surjection $G\to G/Z_i$, giving us 
\[
Z_0 \nor Z_1 \nor Z_2 \nor \cdots.
\]
Set $C_0=G$ and inductively define $C_{i+1}=[C_i,G]$. This gives us
\[
C_0 \nnor C_1 \nnor C_2 \nnor \cdots.
\]
\begin{enumerate}
\item If $Z_k=G$ for some integer $k$, show that $C_i \subseteq Z_{k-i}$ for all $1 \le i \le k$.
\item If $C_k=\{ e\}$ for some integer $k$, show that $C_{k-i} \subseteq Z_i$ for all $1 \le i \le k$.
\end{enumerate}
Hence $Z_k=G$ if and only if $C_k=\{ e\}$. In this case, $G$ is \emph{nilpotent}; the least $k$ is the \emph{nilpotency class} of $G$.

%\textbf{Solution:} 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item Let $G$ be a nilpotent group, and let $H$ be the set of elements of finite order.
\begin{enumerate}
\item Prove that $H$ is a subgroup of $G$. \quad Hint: Use induction on the nilpotency class of $G$.
\item Prove that every finite subset of $H$ generates a finite subgroup.
\item Prove that elements of $H$ of relatively prime order commute.
\end{enumerate}

%\textbf{Solution:} 
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item Let $G$ be the group of upper triangular matrices $\begin{pmatrix}a & c\\0 & b\\ \end{pmatrix}$ in $\GL_2(\RR)$. Show that $G$ is solvable. Is it nilpotent? 

%\textbf{Solution:} 

\end{enumerate}
\end{document}

