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\pagestyle{myheadings} \markright{Math 3340 (Spring 2021) \hspace{2in}
(HW4)}

\begin{document}


%% PREAMBLE

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%\vspace{.05in}
  {\large Homework \# 4 $\quad$ Math 3340 Spring 2021

    \medskip
    
    Due at the end of the day Monday, 15 Mar, at midnight.
  }

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Please submit your completed homework via gradescope on canvas.
I encourage you to work with your classmates on this homework (except for the journal entries!)
When you submit your work, please {\bf list your collaborators}.
(Your grade will not be affected.)
Even if you work in a group, you should write up 
your solutions {\bf yourself}! 
You should include all computational details, and proofs should be carefully 
written with full details.
As always, please write {\bf neatly and legibly} (feel free to use \LaTeX\ to write up your solutions, if you wish!).

\vskip 4mm

%% END PREAMBLE

%%   \noindent {\bf Reading.} {\em The Secret to Raising Smart Kids} by Carol Dweck, in Scientific American.  You can find it
%% by Google, or in the files section on our canvas site (it is also linked to from the syllabus).

%% \vskip 0.1in

\noindent {\bf Journal entry.} There is no journal entry this week.

\vskip 0.1in

\noindent {\bf Exercises.} 

\vskip 0.1in

\begin{enumerate}


\item Let $G$ be a finite group, and suppose that $H$ has index 2 in $G$.  Show that $H$ is a normal subgroup.
  
\item Suppose that $G$ is a finite group.  For $g \in G$, define $\rho_g : G \rightarrow G$
  by $\rho_g(a) := ga$.
  \babc
\item Show that for any $g$, $\rho_g$ is a bijection, and therefore an element of $Sym(G)$, or, in other words,
  is a permutation on the elements of $G$.
\item Show that for all $g, h \in G$, $\rho_{gh} = \rho_g \circ \rho_h$.
\item Define a function $\phi : G \to Sym(G)$ by setting $\phi(g) := \rho_g$.
\item Show that $\phi$ is a group homomorphism.
\item Find the kernel of $\phi$.
  \item Prove that if $G$ has $n$ elements, then $G$ is isomorphic to a subgroup of $S_n$.
  \eabc

\item Let $G$ be a group.  An automorphism of $G$ is an isomorphism $f : G \to G$.  The set of all automorphisms
  of $G$ is denoted by $Aut(G)$.
  \babc
\item Show that $Aut(G)$ is a group.
\item Show that for $g \in G$, the conjugation map $\tau_g : G \to G$, defined by $\tau_g(a) := g a g^{-1}$
  is an automorphism of $G$.
\item Define a function $\phi: G \to Aut(G)$ by $\phi(g) := \tau_g$.  Show that $\phi$ is a homomorphism.
  \item What is the kernel of $\phi$?
\item Find $Aut(\Z_3)$.
\item Find $Aut(S_3)$.
%% \item Find $Aut(D_4)$.
  \eabc

%% \item Find all of the subgroups of $S_3$.  Which of these subgroups is normal?

\item Find all of the subgroups of $D_4$.  Determine which of these subgroups is normal.

  \item In this problem, we consider even and odd permutations.
    A permutation $\sigma \in S_n$ is called ${\bf even}$ if it can be written as the product of an even number of transpositions,
    and it is called {\bf odd} if it can be written as the product of an odd number of transpositions (so at the moment,
    it is possible that a permutation is both even and odd, but we prove in part (d) that this cannot happen).
  A {\bf complete factorization} of $\sigma \in S_n$ is a representation of $\sigma$
  as a product of disjoint cycles, where all 1-cycles are represented explicitly.
  For example, the complete factorization of
  $\sigma = \begin{bmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 \\
                    3 & 2 & 7 & 4 & 6 & 5 & 1
  \end{bmatrix}$
  is $(1\, 3\, 7)(2)(4)(5\, 6)$. In this problem, you may use the fact
  that the complete factorization of a permutation $\sigma$ is unique,
  up to order.

  Define the {\bf sign} of a permutation $\sigma \in S_n$ as follows:
  write the complete factorization of $\sigma$ to be: $\sigma = \sigma_1 \sigma_2 \ldots \sigma_m$.
  Define $sign(\sigma) = (-1)^{n-m}$.  For example, the sign of the above permutation is $(-1)^{7 - 4} = -1$.
  This permutation can be written as a product of 3 transpositions.  But that doesn't (yet) make it odd,
  since perhaps there is another way of writing it as a product of an even number of transpositions.
  In this problem, we show that this cannot happen, and understand better what even and odd permutations
  are.
  \babc
\item Given $k \ge 1$, what is the sign of a $k$-cycle in $S_n$?
\item Suppose that $k, \ell \ge 0$ and that $a, b, c_i, d_j$ are distinct integers in $[n]$.  Show that
  \[ (a\, b)(a\, c_1\, \ldots\, c_k\, b\, d_1\, \ldots\, d_\ell) = (a\, c_1\, \ldots\, c_k) (b\, d_1\, \ldots\, d_\ell)\, \]
and
\[ (a\, b)(a\, c_1\, \ldots\, c_k) (b\, d_1\, \ldots\, d_\ell) = (a\, c_1\, \ldots\, c_k\, b\, d_1\, \ldots\, d_\ell) \]
\item Show that if $\tau$ is a transposition, and $\sigma \in S_n$, then
  $sign(\tau \sigma) = - sign(\sigma)$.
\item Show that if a permutation $\sigma \in S_n$ can be written as a product of an odd number of transpositions,
  then it can't be written as a product of an even number of transpositions.  Conclude that every permutation
  is either even or odd, but not both.
\item Show that the function $sign: S_n \longrightarrow \{1, -1\}$, where $\{1, -1\}$ is a group under multiplication,
  is a group homomorphism.
\item Let $A_n$ be the set of all even permutations in $S_n$.  Show that it is a subgroup of $S_n$, and that
  this subgroup is normal.
  \eabc

%% \item Consider the subgroup $H$ of $S_4$ generated by the permutations $(1,2)(3,4)$ and $(1,4)$.
%%   {\bf This one is a holdover, keep it?  Modify it?  Dump it?}
%%   \babc
%% \item What is the order of this subgroup?
%% \item Find the left cosets of $H$ in $S_4$.
%% \item Find the right cosets of $H$ in $S_4$.
%% \item Find the center $Z(H)$ of $H$.
%% \eabc

\end{enumerate}
\end{document}

