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\begin{document}

\textbf{Math 6310, Assignment 3 \hfill Due in class: 19 October, Monday}

% \medskip {\large Name: }

\begin{enumerate}

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\item Let $n \ge 3$. Prove that $x^3=(1\ 2\ 3)$ has no solution in $S_n$.

%\textbf{Solution:} 

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\item How many elements of the group $S_8$ commute with the permutation $(12)(34)(56)$\,?

%\textbf{Solution:} 
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\item Let $H(\FF_{p})$ be the Heisenberg group over $\FF_{p}$ constructed in class as the semidirect product $(\ZZ/p \times \ZZ/p)\rtimes \ZZ/p$, where the automorphism is given by the matrix
\[
\begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix} \in \GL_{2}(\FF_{p})
\]
Prove that if $p$ is odd, then $H(\FF_{p})$ has exponent $p$, and that $H(\FF_{2})$ is isomorphic to $D_{4}$, and so of exponent $4$.
 
%\textbf{Solution:} 
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\item 
Let $H$ be a group, $K$ a finite cyclic group and $\varphi_{1},\varphi_{2}\colon K\to \Aut(H)$ homomorphism of groups.
Prove that if $\varphi_{1}(K)$ and $\varphi_{2}(K)$ are conjugate subgroups of $\Aut(H)$, then there is an isomorphism of groups
\[
H\rtimes_{\varphi_{1}} K \cong H\rtimes_{\varphi_{2}} K
\]
Hint: Suppose $\sigma \varphi_{1}(K) \sigma^{-1} = \varphi_{2}(K)$ for some $\sigma$ in $\Aut(H)$, then there exists an integer $a$ such that
\[
\sigma \varphi_{1}(k) \sigma^{-1} = \varphi_{2}(k)^{a}\quad\text{for all $k\in K$.}
\]
The map $H\rtimes_{\varphi_{1}} K \to H\rtimes_{\varphi_{2}} K$ defined by $(h,k)\mapsto (\sigma(h),k^{a})$ is the desired homomorphism.

%\textbf{Solution:} 
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\item Let $G$ be a finite group, $K$ a normal subgroup of $G$, and $P$ a Sylow $p$-subgroup of $K$. Prove that $G=KN_P$, where~$N_P$ is the normalizer of $P$ in $G$.

%\textbf{Solution:} 
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\item Determine, up to isomorphism, all groups of order $99$.

%\textbf{Solution:} 
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\item Determine, up to isomorphism, all groups of order $63$.

%\textbf{Solution:} 
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\item Let $G$ be a finite group, and $\phi\colon G\to G$ a homomorphism.
\begin{enumerate}
\item Prove that there exists $n$ such that $\Image\phi^m=\Image\phi^n$ and $\ker\phi^m=\ker\phi^n$ for all $m\ge n$.
\item For $n$ as above, prove that $G$ is the semidirect product $(\ker\phi^n)\rtimes(\Image\phi^n)$.
\end{enumerate}

%\textbf{Solution:} 
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\item Let $G$ be a finite group, and $p$ a prime dividing $|G|$ such that the map $x\mapsto x^p$ is a homomorphism.
\begin{enumerate}
\item Prove that $G$ has a unique Sylow $p$-subgroup $P$.
\item Prove that there exists $N\nor G$ such that $N\cap P=\{e\}$ and $G=PN$.
\item Show that $G$ has a nontrivial center.
\end{enumerate}

%\textbf{Solution:} 
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\item Let $|G|=p^km$ where $p$ is a prime. Let $X$ be the set of $p^k$-element subsets of $G$.
\begin{enumerate}
\item Show that $|X|/m \equiv 1 \mod p$.

\item Let $G$ act on $X$ by left translation, i.e., $g(S)=gS$ for $S\in X$. Prove that the order of each stabilizer subgroup~$G_S$ divides $p^k$. (Hint: $G_S$ acts on $S$ by left translation.)

\item Let $Y=\{S \in X : |G_S|=p^k\}$, and show that $|X|\equiv |Y| \mod pm$.

\item Prove that $Y=\{Hx : H \text{ is a subgroup of $G$ with }|H| = p^k, \text{ and } x \in G \}$.

\item Conclude that the number of subgroups of $G$ of order $p^k$ is $1 \mod p$.
\end{enumerate}
This extends the Sylow theorems, since we did not assume $m$ is relatively prime to $p$.

%\textbf{Solution:} 
\end{enumerate}
\end{document}

