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\begin{center}
  {\large Mathematics 101}\\Fall 2014\\Homework 8 (not to hand in)
\end{center}

\begin{enumerate}
\item Let $p$ be an odd prime, $n \ge 2$, and let $G = GL_n(\F_p)$,
  where $\F_p = \Z/p\Z$ is the field with $p$ elements.
  \begin{enumerate}
  \item Determine the size of $G$ and via the first isomorphism
    theorem, deduce the order of the subgroup $H = SL_n(\F_p)$.

  \item Show that $GL_n(\F_p)$ is a split extension of $SL_n(\F_p)$ by
    $\F_p^\times$.

\item Show that every Sylow $p$-subgroup of $G$ is also a Sylow
  $p$-subgroup of $H$.



\item  Now restrict $n$ to be $n = 2$ or $n = 3$.  For each case
  exhibit a Sylow $p$ subgroup.  \textit{Hint:} If the Sylow subgroups
  have size $p^m$, then every element of a Sylow $p$-subgroup in
  annihilated by the polynomial $x^{p^m} - 1 \in \F_p[x]$.  Note that you
  are in characteristic $p$, and you are dealing with matrices, so
  perhaps a little linear algebra might apply.

\item For each of $n = 2, 3$ determine how many Sylow $p$-subgroups
  there are in $GL_n(\F_p)$.


  \end{enumerate}

\item Semidirect products.  We showed in class that $Aut(\Z_n) \cong
  \Z_n^\times$.

  \begin{enumerate}


  \item Suppose that $H_1$, $H_2$ and $K$ are groups, $\sigma: H_1 \to
    H_2$ is an isomorphism, and \newline $\psi:H_2 \to Aut(K)$ a
    homomorphism, so that $\varphi = \psi\circ \sigma: H_1 \to Aut(K)$
    is also a homomorphism.  Show that $K \rtimes_\varphi H_1 \cong K
    \rtimes_\psi H_2$.
  
  \item Suppose that $H$ and $K$ are groups and $\varphi, \psi:H \to
    Aut(K)$ are monomorphisms with the same image in $Aut(K)$.  Show
    that there exists a $\sigma \in Aut(H)$ such that $\psi =
    \varphi\circ \sigma$.

  \item Suppose that $H$ and $K$ are groups, $\varphi, \psi:H \to
    Aut(K)$ are monomorphisms, and $Aut(K)$ is finite and cyclic.
    Show that $\varphi$ and $\psi$ have the same image in $Aut(K)$.

  \item Let $p < q$ be primes with $p \mid (q-1)$. Let $H$ and $K$ be
    cyclic groups of order $p$ and $q$ respectively.  Let $\varphi,
    \psi: H \to Aut(K)$ be nontrivial homomorphisms. Observing that
    $Aut(K)$ is cyclic, show that $K\rtimes_\varphi H \cong
    K\rtimes_\psi H$.

  \item Let $p < q$ be primes. Show that up to isomorphism, there are
    at most two groups of order $pq$.
  \end{enumerate}

\item Let $H$ be a group, and by $H^n$ denote group which is the
  external direct product of $H$ with itself $n$ times.  Show that the
  symmetric group $S_n$ acts on $H^n$ via $(\sigma, (h_1, \dots, h_n))
  \mapsto (h_{\sigma^{-1}(1)}, \dots, h_{\sigma^{-1}(n)})$.  Note that
  the obvious map $(\sigma, (h_1, \dots, h_n)) \mapsto (h_{\sigma(1)},
  \dots, h_{\sigma(n)})$ is a right action, that is you need the
  inverses so that the permutation representation is a homomorphism.

  \textit{Hint:} This is a bit subtle with notation; you may find the
  following observation useful.  In set theory, $Y^X$ denotes the set
  of functions $f: X\to Y$, so one can interpret $H^n$ as $H^X$ where
  $X = \{1, 2, \dots, n\}$.  Now show that the natural action of $S_n$
  on $X$ induces an action of $S_n$ on $H^X \cong H^n$ as suggested
  above.

\item Let $A$, $B$ be groups and let $|B|=n$.  We know that $B$ acting
  on itself by left translation induces an injective homomorphism
  $\rho:B \to S_n$; this permutation representation is usually called
  the \textit{left regular representation}.  As we saw in the previous
  problem, $S_n$ acts on $A^n$ with permutation representation
  $\varphi: S_n \to Aut(A^n)$.  The composition of $\varphi$ and
  $\rho$, is a homomorphism $\varphi\circ \rho: B \to Aut(A^n)$.  The
  \textit{wreath product} of $A$ by $B$, denoted $A \wr B$, is defined
  to be the semidirect product $A\wr B = A^n
  \rtimes_{\varphi\circ\rho} B$.

  Show (obvious) that the order of $A\wr B$ is $|A|^{|B|}\cdot |B|$.
  From this it is clear that $\Z_2\wr \Z_2$ is a group of order 8.
  Determine its isomorphism class in part by writing out all eight
  elements and finding their orders.


\item The point of this exercise is to show that for $p$ a prime, any
  Sylow $p$-subgroup of $S_{p^2}$ is isomorphic to $\Z_p \wr \Z_p$.
  \begin{enumerate}
  \item First compute the cardinality of a Sylow $p$-subgroup of
    $S_{p^2}$ (without assuming the isomorphism).

  \item The following is outline of a proof is from Rotman's
    \textit{Theory of Groups}, where a more general result is
    established.  Let $B_0 = \{1, 2, \dots, p\}$ and let $B_i = B_0 +
    ip$, so that the set $\{1, 2, \dots, p^2\}$ is the disjoint union
    $B_0\cup \cdots \cup B_{p-1}$.  Consider the permutation $\sigma
    \in S_{p^2}$ which for $b_0 \in B_0$ takes
    \[
    \sigma(b_0 + ip) =
    \begin{cases}
      b_0 + (i+1)p& \textrm{if $i < p-1$,}\\b_0&\textrm{if $i = p-1$}.
    \end{cases}
    \]

    Verify that $B_{i+1} = \sigma(B_i)$ (with the subscripts read
    modulo $p$), and that $\sigma$ has order $p$.

  \item Let $H = H_0$ denote a Sylow $p$-subgroup of $S_p$; observe
    that it is cyclic of order $p$.  Viewing $H_0$ also a subgroup of
    $S_{p^2}$, let $H_1$ be the image of $H_0$ under the action of
    $\sigma$; so for example the cycle $(1\, 2\, 3 \dots p)$ would go
    to $(p+1\ p+2\ \dots 2p)$.  Now let $H_{i+1} = \sigma(H_i)$.  Show
    that $S_{p^2}$ contains a subgroup $K = H_0\cdots H_{p-1} \cong
    H_0 \times \cdots \times H_{p-1}\cong H^{p}$. \textit{Hint:
      Disjoint permutations commute.}

  \item Show that $K \cap \langle \sigma \rangle = \{e\}$.


  \item Now the tricky bit.  Rotman wants you to show first that
    conjugating by $\sigma$ maps from
    $H_0 \times \cdots \times H_{p-1}$ to
    $H_1\times H_2\times \cdots \times H_{p-1}\times H_0$. That part
    is easy.  Then he wants you to show that the induced action on
    $H^p = H_0^p$ is the one you need to identify
    $\langle K, \sigma\rangle$ as a wreath product.  The subtlety is that
    $H^p \cong H_0\times \cdots \times H_{p-1}$ and you need to chase
    through the identifications to verify his claim.

  \item Conclude that $\langle K, \sigma\rangle \cong H \wr \langle
    \sigma \rangle \cong \Z_p \wr \Z_p$
  \end{enumerate}

\end{enumerate}

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