\documentclass[12pt]{amsart}
\usepackage{amsthm,amsopn,amsfonts,amssymb}
\usepackage{hyperref,cancel}%,stmaryrd}
%\usepackage{tabmac}
%\usepackage[all]{xy}
% ^C ^T ^P to put AucTeX in Tex-PDF mode
\usepackage{fancyhdr}
\usepackage[dvips]{graphics}
\usepackage{young}

\usepackage{euler, amsfonts, amssymb, latexsym, epsfig}
\usepackage{epstopdf}

\renewcommand{\familydefault}{ppl}

\setlength{\oddsidemargin}{0in} \setlength{\evensidemargin}{0in}
\setlength{\marginparwidth}{0in} \setlength{\marginparsep}{0in}
\setlength{\marginparpush}{0in} \setlength{\topmargin}{-.5in}
\setlength{\headheight}{6pt} \setlength{\headsep}{20pt}
\setlength{\footskip}{.3in} \setlength{\textheight}{9.2in}
\setlength{\textwidth}{6.5in} \setlength{\parskip}{4pt}


\newtheorem{Theorem}{Theorem}[section]
\newtheorem{thm}[Theorem]{Theorem}
\newtheorem*{Theorem*}{Theorem}
\newtheorem{Lemma}[Theorem]{Lemma}
\newtheorem{lem}[Theorem]{Lemma}
\newtheorem{Proposition}[Theorem]{Proposition}
\newtheorem{prop}[Theorem]{Proposition}

\newtheorem{conjecture}[Theorem]{Conjecture}
%\newtheorem{formula}[Theorem]{Formula}
\newtheorem{Corollary}[Theorem]{Corollary}
\newtheorem{cor}[Theorem]{Corollary}

\theoremstyle{definition}
\newtheorem{definition}[Theorem]{Definition}


\newtheorem{property}[Theorem]{Property}

\theoremstyle{remark}
\newtheorem{example}[Theorem]{Example}
\newtheorem{remark}[Theorem]{Remark}
\newcommand{\Xo}{\mathring{X}}
\newcommand{\Pio}{\mathring{\Pi}}
\newcommand{\SR}{\mathrm{SR}}
\newcommand{\In}{\mathrm{In}}
\newcommand{\AFl}{\widetilde{{\mathcal F} \ell}}
\newcommand{\AGr}{{\mathcal G}r}
\newcommand{\B}{{\mathcal B}}
\newcommand{\C}{{\mathbb C}}
%\newcommand{\Gr}{\mathrm{Gr}}
%\newcommand{\Grkn}{\Gr_k(\AA^n)}
\newcommand{\GS}{S_{n,k}^{\mathrm{min}}}
\newcommand{\AGS}{S_{n,k}^{\mathrm{max}}}
\newcommand{\I}{{\mathcal I}}
\newcommand{\JJ}{G}
\newcommand{\J}{{\mathcal J}}
\newcommand{\M}{{\mathcal M}}
\newcommand{\pt}{\mathrm{pt}}
\newcommand{\Q}{{\mathcal Q}}
\newcommand{\R}{{\mathbb R}}
\newcommand{\tF}{\tilde F}
\newcommand{\tsig}{\tilde \sigma}
\newcommand{\tS}{\tilde S}
\newcommand{\weak}{\mathrm{weak}}
\newcommand{\Z}{\mathbb Z}
\newcommand{\from}{\leftarrow}
\newcommand{\onto}{\twoheadrightarrow}
\newcommand{\fromonto}{\twoheadleftarrow}
\newcommand{\into}{\hookrightarrow}
\newcommand\mapsfrom{\reflectbox{$\mapsto$}}

\newcommand{\CC}{\mathbb{C}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\PP}{\mathbb{P}}

\newcommand\tensor{\otimes}
\newcommand\nwsearrow{\nwarrow \hskip -.16in \searrow}

\newcommand{\fixit}[1]{\texttt{*** #1 ***}}

\def\acts{\hspace{.1cm}{
    \setlength{\unitlength}{.25mm}
    \linethickness{.08mm}
    \begin{picture}(8,8)(0,0)
    \qbezier(7,6)(4.5,8.3)(2,7)
    \qbezier(2,7)(-1.5,4)(2,1)
    \qbezier(2,1)(4.5,-.3)(7,2)
    \qbezier(7,6)(6.1,7.5)(6.8,9)
    \qbezier(7,6)(5,6.1)(4.2,4.4)
    \end{picture}\hspace{.1cm}
    }}
\def\acted{\hspace{.1cm}{
    \setlength{\unitlength}{.25mm}
    \linethickness{.08mm}
    \begin{picture}(8,8)(0,0)
    \qbezier(1,6)(3.5,8.3)(6,7)
    \qbezier(6,7)(9.5,4)(6,1)
    \qbezier(6,1)(3.5,-.3)(1,2)
    \qbezier(1,6)(1.9,7.5)(1.2,9)
    \qbezier(1,6)(3,6.1)(3.8,4.4)
    \end{picture}\hspace{.1cm}
    }}

\newcommand\defn[1]{{\bf #1}}
\newcommand\yy{{\bf y}}
\newcommand\iso{\cong}
\newcommand\shift{\textcyr{X}}
\newcommand\sweep{\Psi}
\newcommand\St{{\rm St}}
\newcommand\dom{\,\backslash\,}
\newcommand\barX{{\overline X}}
\newcommand\integers{\ZZ}
\newcommand\codim{{\rm codim\space }}
\newcommand\calP{{\mathcal P}}
\newcommand\naturals{{\mathbb N}}
\newcommand\Kint{K\!\!\!\!\!\int\ }
\newcommand\calO{{\mathcal O}}
\newcommand\dash[1]{-\!\!\! #1\!\!\!-}
\newcommand\bslash{\big\backslash\!\!\!}
\newcommand\fslash{\big /\!\!\!}
\newcommand\union{\cup}
\newcommand\Union{\bigcup}

\newcommand\complexes{{\mathbb C}}
\newcommand\comment[1]{{\bf *** #1 *** }}
\newcommand\Gr[2]{Gr_{#1}(\AA^{#2})}
\newcommand\Grkn{\Gr{k}{n}}
\newcommand\wt[1]{{\widetilde #1}}
\newcommand\Gm{{{\mathbb G}_m}}


\newcommand\rank{{\rm rank}}
\newcommand\diag{{\rm diag}}
\newcommand\rowspan{{\rm rowspan}}

\DeclareMathOperator{\Spec}{Spec}
\DeclareMathOperator{\PosSpan}{\reals_+}
\DeclareMathOperator{\av}{av} \DeclareMathOperator{\Par}{Par}
\DeclareMathOperator{\Bound}{Bound}
\DeclareMathOperator{\Neck}{Neck} \DeclareMathOperator{\GGMS}{GGMS}
\DeclareMathOperator{\Jugg}{Jugg} \DeclareMathOperator{\Span}{Span}
\newcommand{\Fl}{F \ell}
\renewcommand\O{{\mathcal O}}
\renewcommand\AA{{\mathbb A}}
\newcommand\junk[1]{}
\newcommand\optional[1]{\fixit{I may cut the following:}\hrule #1 \hrule}
\newcommand\rationals{{\mathbb Q}}
\newcommand\reals{{\mathbb R}}

\pagestyle{empty}

\begin{document}

\title{Math 4340 final, Spring 2020}

\author{due 11:59pm Friday May 22, Eastern time}

\maketitle

Once you look at p2 of this exam, you may only make reference to your
class notes and to Dummit \& Foote, no other sources (books, web pages,
people, etc.)

I'd {\em prefer} if your final were composed in \LaTeX,
and promise you that (1) if you're not yet good at \LaTeX\ that as a
math person you will need to get there, 
and (2) the time spent on learning that for this final will be minuscule
on top of actually doing the math. I have put the source for this
final exam on the Canvas page by way of random example,
so if you've never \TeX ed before just
copy that \LaTeX\ file, skip past the preamble, change the {\tt
  $\backslash$title}, and edit in between the {\tt
  $\backslash$maketitle} and $\backslash${\tt end$\{$document$\}$}.
If you need to find commands for symbols, Google {\tt detexify}. 
All that said, if you still aren't willing to try \LaTeX,
I will accept alternate formats, e.g. scans of handwritten answers.

In each answer, write as much as it takes to convey your thought
process; full English sentences are much easier to give credit to than
bare, unmotivated scribbled formul\ae.  (They won't do any good if
they can't be read, so {\em do} put effort into making them legible.)

If two questions have the same number, they concern the same object.
For example, the relation in 4b is the one defined in 4. [$5\pi$]
indicates that a problem is worth $5\pi$ points.

{\em Feel free} to ask me questions (by email) during the test,
especially if you need a little reminder about a definition. Worst
case is I don't answer.  (It's very sad to afterward hear ``I didn't
realize I could ask you that'' ---\, find out!) Likewise, check back
on the Canvas page to see if there are updates to the questions.

%\markright{\MakeUppercase{ }}

\parindent=0cm

% . subspaces
% matrix mult
% det, cross product, dot product, perp
% . open, closed, continuity
% . derivatives

% \vskip .1in \hrule \vskip .1in

\newcommand\pbr{\vfill\eject . \vfill\eject \addtocounter{page}{-1}}
\renewcommand\pbr{\vskip .1in \hrule \vskip .1in}

\vfill

{\bf Turn over only when you're ready to foresake all other sources
  except for your class notes and Dummit \& Foote...}

\vfill

\eject

\newcommand\FF{{\mathbb F}}

1. Let $G \leq GL(4,\CC)$ be the set of upper triangular matrices
$$
\begin{bmatrix}
  a & b & c & d \\ 0  & e & f & g \\ 0 & 0 & h & i \\ 0&0&0& k 
\end{bmatrix}
$$

1a [2]. What are the conditions on $a,\ldots,k$ such that this
matrix is invertible?

1b [3]. Let $T \leq G$ be the subgroup of diagonal matrices. \\
Find another subgroup $F \trianglelefteq G$ such that $G$
is a semidirect product $T \ltimes F$. 

1c [5]. Find at least three other ways to see $G$ as a semidirect product.

\pbr

2 [5]. Let $G$ be a potentially infinite group, but
$H \leq G, H \not\trianglelefteq G$ an abnormal subgroup with $G/H$
finite. Show that $H$ contains a proper\footnote{meaning, $K\neq H$}
normal subgroup $K \triangleleft H$ of finite index.

\pbr

3 [10]. How many automorphisms does the group $\ZZ_{p^2} \times \ZZ_{p^2}$ have?

\pbr

4. For
$\lambda = (\lambda_1\geq \lambda_2 \geq \ldots \geq \lambda_k) \in \naturals^k$,
and $p$ prime, let $G_\lambda$ denote the finite abelian group
$\prod_{i=1}^k \ZZ_{\left( p^{\lambda_i}\right)}$, e.g.
$G_{3,1} = \ZZ_{p^3} \times \ZZ_{p}$.

Consider ``short exact sequences''
$$ 1 \to A \to B \to C \to 1 $$
of groups, meaning, $A$ is (corresponded with) a normal subgroup of $B$
and $C$ with the quotient group $B/A$.

4a [3]. If $1 \to G_\lambda \to G_\mu \to G_\nu \to 1$ is a
short exact sequence, show that $|\mu| = |\lambda| + |\nu|$.

4a [7]. If $1 \to G_\lambda \to G_\mu \to G_\nu \to 1$ is a
short exact sequence, show that $\mu_1 \leq \lambda_1 + \nu_1$.

Fun fact (you may not use!): there exact such sequences iff
there exist Hermitian matrices $H_\lambda,H_\mu,H_\nu$ with those
natural numbers as eigenvalues, and $H_\mu = H_\lambda + H_\nu$.

\pbr

5. Let $GL_n(\FF_p)$ be the group of $n\times n$ invertible matrices
over $\FF_p$. If you find this problem hard, try $n=2$.

5a [3]. Looking one row at a time, show that
$$ \# GL_n(\FF_p) = (p^n-p^0)(p^n-p^1)(p^n-p^2)\cdots (p^n-p^{n-1}) $$

5b [2]. How big are its Sylow $p$-subgroups?

5c [5]. Find a Sylow $p$-subgroup of $GL_n(\FF_p)$.

5d [5]. How many $p$-Sylow subgroups are there?

\pbr

6 [10]. Let $G$ be the group of $3\times 3$ complex matrices, where each
row and column has two zeros and a $\pm 1$, e.g.
$\begin{bmatrix}  0 & 1 & 0 \\ 0 & 0 & -1 \\ -1 & 0 & 0 \end{bmatrix}$.
Describe the conjugacy classes in this group -- how many are there,
of what sizes. (You may use a computer for this question, if that helps.)

\pbr

7. Let $\FF_4 = \{0,1,a,a+1\}$ where $a^2 = a+1$.\\
7a [10]. Find an irreducible quadratic polynomial $p(x) \in \FF_4[x]$.
(Prove it's irreducible.)

7b [5]. Find
a generator of the multiplicative group of $\FF_4[x]/\langle p(x)\rangle$.

\pbr

8 [10]. Let $p(x) = (x^2-2)^2 - 3 \in \QQ[x]$. Find its splitting field $E$,
its Galois group $Gal(E,\QQ)$, and all intermediate subfields.

\pbr

9 [5]. Let $G$ be finite, and $V$ a finite-dimensional complex
representation over $\CC$.
Show that $V^{\tensor n} := V \tensor V \tensor\cdots\tensor V$
is irreducible for all $n>0$ iff $V$ is $1$-dimensional.
(Recall our definition of $V\tensor W$ for column vectors,
considered as $k\times 1$ matrices, using the Kronecker product of matrices.)

\pbr

10. A mysterious group $G$ has the following partial character table
(one column for each conjugacy class, one row for each character of irrep,
possibly in nontraditional order):
$$
\begin{matrix}
        && I &  II & III & IV \\ \\
 \chi_1 && 1  & ???& 1 & 1 \\
 \chi_2 && 1  &  1 & ??? & ??? \\
 \chi_3 && ???&  1 & \omega & ??? \\
 \chi_4 && -1 &  3 & ??? & 0
\end{matrix}
$$

10a [2]. Which column is that of the identity element (and why)?

10b [2]. What is the order of the group (and why)?

10c [4]. How big is conjugacy class I (and why)?

10d [2]. How big are the other conjugacy classes (and why)?

If along the way you figure out some entries in the table, say so.

\end{document}

