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\begin{center}
{\large\textbf{Dartmouth College}\\
Mathematics 101}
\end{center}
\begin{center}
Assignment 1 (aka, Awakening from hibernation):\\
due Thursday, October 1
\end{center}
It is assumed you have read Chapter 0 and Chapter 1.1 - 1.6 as well as
the few sections we are covering for this week in Chapter 2.  The
following set of exercises is meant to reengage your brains by having
you play with some explicit examples and by helping you recall
some basic facts from your undergraduate linear and abstract algebra
courses.

\begin{enumerate}
\item Let $\sigma = (a_1 a_2 \dots a_m)$ be an $m$-cycle in
  the symmetric group $S_n$ ($n \ge m$).  
  \begin{enumerate}
  \item  (p33, \#10) Show that for $1 \le i,k \le
  m$, $\sigma^i(a_k) = a_{k+i}$ where the subscript $k+i$ is
  interpreted as the least positive residue of $k+i$ modulo $m$.
  Deduce that the order of $\sigma$ is $m$.

\item (p33, \#11 modified) Show that $\sigma^i$ is an $m$-cycle if and
  only if $\gcd(i,m) = 1$.
  \end{enumerate}



\item (p33, \#15) Prove that the order of an element in $S_n$ equals
  the least common multiple of the lengths of its cycles in its cycle
  decomposition.  Hint:  See also exercise 24, in 1.1.

% \item (p35, \#8) Show that for any field $F$, the group $GL_n(F)$ is
%   non-abelian for any $n \ge 2$.

\item (p35) Establish the formula at the end of section 1.4:  Let $\Fq$
  be a finite field with $q$ elements.  Show that
$|GL_n(\Fq)| = (q^n - 1)(q^n - q) \cdots (q^n - q^{n-1})$.   Hint:
Recall that a matrix is invertible if and only if its rows (resp.
columns) are linearly independent.

\item (p40, \#4) Show that the multiplicative groups $\R^\times$ and
  $\C^\times$ are not isomorphic.

% \item (p40, \#19 modified) Let $G = \{ z\in \C \mid z^n = 1\
%   \textrm{for some } n \in \Z_+\}$.  Show that for a fixed positive
%   integer $k$, the map $\varphi: G \to G$ given by $\varphi(z) = z^k$
%   is a surjective homomorphism, but not an isomorphism.  For any $z
%   \in G$, determine the fiber of $\varphi$ over~$z$.



\item (p41, \#25) Let $n$ be a positive integer and let $r,s$ be the
  canonical generators of the dihedral group $D_{2n}$, and let $\theta
  = 2\pi/n$. 
  \begin{enumerate}
  \item Prove that the matrix $\rot$ is the matrix of the linear
    transformation (with respect to the standard basis of $\R^2$)
    which describes a counterclockwise rotation about the origin of
    $\theta$ radians.
  \item Prove that the map $\varphi:D_{2n} \to GL_2(\R)$ defined on
    generators by $\varphi(r) = \rot$ and $\varphi(s) = 
    \spmatrix{0} {1} {1} {0}$ extends to an injective group
    homomorphism.

  \end{enumerate}

\item Let $G = GL_2(F)$ for a field $F$. Let $H$ be the subgroup of
  upper triangular matrices in $G$ (you may assume $H$ is a group).
  \begin{enumerate}
  \item If $F = \C$, show that $G = \bigcup_{g \in G} gHg^{-1}$, that
    $G$ is a union of conjugates of $H$.  Hint: This is really a
    linear algebra problem.  Let $T: \C^2 \to \C^2$ be any linear
    transformation.  Show that there exists a basis $\B$ for $\C^2$ so
    that the matrix of $T$ with respect to $\B$ is upper triangular.
    Hint for hint: think about eigenvectors.
\item  If $F = \R$, does the result remain valid?  Proof or counterexample.
  \end{enumerate}



  \end{enumerate}
%\end{enumerate}


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