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\prof{Peter Cholak and Juan Migliore} \class{Math 222 - Sample Final}
\quizdate{Monday, April 30} \pagestyle{empty}


\begin{document}

\heading

\noindent The final exam will be Monday, May 7 from 1:45 pm to 3:45
pm in 356 FITZ.\\

\noindent The final exam will cover all the material discussed in class
through Chapter 9 (sections 9.1-9.5 -- we did not cover isomorphisms
of groups in 9.3 but otherwise we covered 9.3) Fair game for the final
are all questions (or similar questions) on the previous exams and
sample exams, homework, and quizzes.  The best way to prepare for the
final is to be sure you can do all problems on this sample final, the
previous two sample exams, previous two exams,
the quizzes and the homework.\\

\noindent The exam will consist of five parts.  Part I: Definitions.
Part II: True/False. Part III: Computational Problems.  Part IV: Known
Proofs.  Part V: an easy unknown proof.  The following are some
examples from the latest material.\\

\noindent Part I:  Definitions.\\

\begin{problem} {\bf Define} the following terms:\\
  a) {\em coset} \\ b) \emph{cyclic group} \\ c) \emph{order of a
    group $G$} \\ d) \emph{order of an element $g$ in $G$}
\end{problem}

\noindent Part II:  True/False.\\

\begin{problem} Answer {\bf True} or {\bf False} - no work
  required:\\
  a)  $\{0,2\}$ is a coset of $(\mathbf{Z}_6, +)$.\\
  b) $D_3$ is cyclic.\\
  c) all groups of order 4 are cyclic. \\
  d) The set of $2 \times 2$ matrices with entries from the reals
  numbers is field. \\
  e) No non-zero $2 \times 2$ matrix is a zero divisor.\\
  f) The polynomial $x^4 +x + 1$ over $\mathbb{Q}$ is reducible (Hint:
  Eisenstein's Criterion).
  \end{problem}

\noindent Part III: Computational Problems.  \\

%\begin{problem}
%  List all elements of $\sqrt[12]{1}$ and determine the orders of all
%  elements.
%\end{problem}

%\begin{problem}
%  Find all the primitive 16th roots of unity in $\mathbb{Z}_{17}$.
%\end{problem}

\begin{problem}
  Be sure to know examples of rings, integral domains, fields, groups,
  etc.
\end{problem}

\begin{problem}
  Find all the possible orders of subgroups of $D_5$.  Give examples
  of subgroups of each of these orders.  Compute the index of each of
  these subgroups in $D_5$.
\end{problem}

\begin{problem}
  Find all the cosets of $\{id, (1 2) (3 4)\}$ inside $A_4$.
\end{problem}


\begin{problem} 
  Does $S_4$ have subgroups of the following orders?  Why or why not?
  2, 3, 4, 5, 6, 7, 8, 9, 10. If so be sure to find a subgroup of that
  order.
\end{problem}

\begin{problem} 
  Does $S_7$ have cyclic subgroups of the following orders?  Why or
  why not?  2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. If so be sure to find
  a cyclic subgroup of that order.
\end{problem}


\begin{problem} 
  Consider the subgroup $H = \{Id,(12),(34),(12)(34)\}$ of $S_4$.
  Is $H$ cyclic?
\end{problem}

\noindent Part IV: Proofs.  There will be two or three \emph{known} proofs.
These will come from the following:\\

\begin{problem}
  Show $1\cdot 2 + 2\cdot 3 + 3\cdot4 + \ldots + n(n+1) =
  \frac{n(n+1)(n+2)}3$, for any $n\in\Bbb{N}$.
\end{problem}

\begin{problem}
  Let $m$, $n$ and $k$ be integers such that $(m,n)=1$ and both $m$
  and $n$ divide $k$.  Then prove that the product $mn$ divides $k$.
\end{problem}

%\begin{problem}
%  Prove that if $P(x)$ and $Q(x)$ are polynomials over a field $F$ of
%  degree $m$ and $n$ respectively then the degree of $P(x)Q(x) = m +
%  n$ while the degree of $P(x) + Q(x) = \operatorname{max}(m,n)$.
%\end{problem}

%\begin{problem} Let $G$ be a group and suppose that $G$ satisfies the
%  following property.  whenever $b\cdot a = a\cdot c$, then $b = c$
%  for $a,b,c \in G$. Prove that $G$ must be abelian.
%\end{problem}

%\begin{problem}
%  Prove that if $n$ is an odd positive integer and $\zeta$ is any
%  primitive $n$th root of unity, then so is $\zeta^2$.  Is this also
%  true for $n$ even?  Why or why not?
%\end{problem}

%\begin{problem}
%  Let $a$ and $b$ be nonzero integers such that $g = (a,b)$.  Prove
%  that $\displaystyle\left(\frac{a}{g},\frac{b}{g}\right)=1$.
%\end{problem}

%\begin{problem} 
%  Let $f: G \to H$ be an isomorphism of groups.  Prove that $f(1_G) =
%  1_H$.
%\end{problem}


%\begin{problem} Let $p$ be a prime number and $G$ be a finite group of
%  order $p$.

%  a) Show that $G$ is cyclic.

%  b) Use part (a) to show that there is only one group of order $p$ up
%  to isomorphism.
%\end{problem}

\begin{problem}
  Show that an unit of a ring divides every non-zero element of the
  ring.
\end{problem}


\begin{problem}
  Let $F = GF(p,P(x))$ where $P(x)$ is irreducible over $\Bbb{Z}_p$.
  Show that for any $a$ in $F$ there is at most one $p$th root of $a$.
\end{problem}

\begin{problem}
  Show if $G$ has order $p^n$ (for $n \geq 1$) then $G$ has an element
  whose order is $p$ ($p$ is prime).
\end{problem}

%\begin{problem} Let $G$ be a finite group and $H \subset G$ be a
%  nonempty subset.  Suppose that $H$ is closed under multiplication.
%  Show that $H$ is in fact a subgroup of $G$.
%\end{problem}

%%\begin{problem}
%%  Let $f: X \to Y$ be a function and $X$ and $Y$ are finite sets of
%%  the same size.  Show that $f$ is onto iff $f$ is 1-1.
%%\end{problem}


%\begin{problem}
%  (Section 9.4, Problem 42.)  Suppose that $H$ is a subgroup of $G$
%  and $g$ is an element of $G$.  Prove that the set $$gHg^{-1} =
%  \{ghg^{-1}: h \in H\}$$
%  is a subgroup of $G$.
%    %and is isomorphic to $H$.
%\end{problem}


\begin{problem}
  Let $R$ be a commutative ring with unity and let $U(R)$ denote the
  set of units of $R$.  Show that $U(R)$ is a group under the
  multipliciation of $R$. (This group is called the group of units of
  $R$.)
\end{problem}

 
\begin{problem}
  Let $G$ be the cyclic group of order $n$ and $d$ be a divisor of
  $n$.  Suppose that $H$ is a subgroup of $G$ of order $d$. Show that
  $H$ consists of all the elements $x$ of $G$ with $o(x)|d$.
\end{problem}

\begin{problem}
  Let $h$ be a function from a group $(G,\cdot)$ to a group
  $(H,\cdot)$ such that for any $a,b \in G$, $h(ab)=h(a)h(b)$.  The
  kernel of $h$ is defined to be $\{g \in G : h(g) = 1_H\}$.  Show
  that the kernel of $h$ is a subgroup of $G$.
\end{problem}

%\begin{problem}
%  Let $G$ be a group of the order $n$, and $H$, $K$ be two subgroups
%  of $G$.
  
%  a) Show that $K\cap H$ is a subgroup of $H$.
  
%  b) If $(aK)\cap H\ne\emptyset$ for some $a\in G$, then $(aK)\cap H$
%  is a coset of $K\cap H$ in $H$.
  
%  c) If the number of elements in $(aK)\cap H$ is $d>0$, then $d|n$.
%\end{problem}

\end{document}