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\def \zz {\mathbb{Z}}
\def \cc {\mathbb{C}}
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\prof{Chris Bendel and Peter Cholak}
\class{Math 222 - Sample Final}
\quizdate{Thursday, May 6}
\pagestyle{empty}

\begin{document}

\heading

\noindent The final exam will be Thursday, May 6 from 1:45 pm to 3:45
pm in 118 DBRT (Prof. Cholak's section) or Nieuwland 127 (Prof. Bendel's
section).\\

\noindent The final exam will cover all the material discussed in class
through Chapter 9.  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, and the quizzes.\\

\noindent The exam will consist of four parts.  Part I: Definitions --
There will be six terms to define, worth 4 points each (24 points in
all).  Part II: True/False -- There will be six true/false questions,
worth 2 points each (12 points in total).  Part III: Computational
Problems -- there will be some number of computational problems for a
total of 70 points.  Part IV Proofs -- There will be three proofs,
worth 15 points each (or 45 points total).  There is one bonus point.
The following are some
examples from the latest material along some other possible problems.\\

\noindent Part I:  Definitions.\\

\begin{problem}(4 points each) {\bf Define} the following terms:\\
a) {\em Isomorphism of groups}.\\
b) {\em Coset}. %final question index of subgroup
\end{problem}

\noindent Part II:  True/False.\\

\begin{problem}(2 points each) Answer {\bf True} or {\bf False} - no work
  required:\\
  a)  The quaternions are isomorphic to $D_4$.\\
  b)  The quaternions are cyclic.
\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} Consider the map $f: (\zz_8,+) \to (\zz_8,+)$ by $f(x) = 4x$.
  Is this an isomorphism of groups? Why or why not? % Consider the set
  %$S = \{x \in \zz_8: f(x) = 0\}$.  How many elements are in $S$?  Is
  %it a subgroup of $(\zz_8,+)$.  Why or why not?  If so, is it normal?
\end{problem}

\begin{problem}
  Identify all the subgroups of $S_3$.
\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}
Let $G = \Big{\{}
\begin{pmatrix}
a & a\\
a & a
\end{pmatrix}:
a \in {\mathbb{R}}^*\Big{\}}$.  $G$ is a group under matrix multiplication.

a) Find the identity element of this group.

b) Find the inverse of $\begin{pmatrix}
3 & 3\\
3 & 3
\end{pmatrix}$ in $G$.
\end{problem}


\begin{problem} The set $G =\{5, 15, 25, 35\}$ is a group under
multiplication modulo 40.

a) Find the identity element in $G$.

b) Find the inverses of the remaining elements in $G$.

c) Which group of order 4 is $G$ isomorphic to?  Why?
\end{problem}

%\begin{problem} Let $G$ be the set of rational numbers {\em except} $-1$. This
%forms a group under the following multiplication: $a*b \equiv a + b + ab$,
%where
%the operations on the right hand side are as usual.

%a) Find the identity element in $G$.

%b) Find the inverse of 2 in $G$.
%\end{problem}

%\begin{problem}
%Let $G$ be a group and $g$ an element of $G$.  We define the {\em
%centralizer of $g$ in $G$} to be
%$$
%Z_g(G) = \{x \in G: xg = gx\}.
%$$
%$Z_g(G)$ is a subgroup of $G$. Compute the following centralizers.

%a) $Z_2((\zz_5,+))$

%b) $Z_{F}(D_4)$

%c) $Z_{(12)}(S_3)$
%\end{problem}

\begin{problem} Let $G$ be a finite group with order 187. 
  Suppose that $H$ and $H'$ are two subgroups of $G$ having the same
  order.  Show that $H$ and $H'$ must be isomorphic.  Hint: Factor
  187.
\end{problem}

%\begin{problem} Compute the Cayley representation
%  of the element $i$ in the quaternions $Q$.
%\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
  cyclic 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}
  Find all the cosets of $\{id, (1 2) (3 4)\}$ inside $A_4$.
\end{problem}

\begin{problem} Which of these maps of groups are isomorphisms? Why?

a) $f: (\cc,+) \to (\cc,+)$ by $f(a + bi) = a - bi$.

b) $f: (\cc^*,\cdot) \to (\cc^*,\cdot)$ by $f(a + bi) = a - bi$.

c) $f: (\mathbb{Q}^*,\cdot) \to (\mathbb{Q}^*,\cdot)$ by $f(x) = 1/x$.

d) $f: (\rr,+) \to (\rr,+)$ by $f(x) = |x|$.
\end{problem}


\begin{problem} Consider the subgroup $H = \{Id,(12),(34),(12)(34)\}$ of $S_4$.
Which of the two groups of order 4 is this isomorphic to. Why?
\end{problem}

\begin{problem}  How many groups of order 167 are there?  Why?  How many
subgroups can a group of order 167 have?
\end{problem}

\noindent Part IV: Proofs.  There will be three proofs, worth 15
points each.  These problems will come from either of the first two
sample exams or the following:\\

%\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} For each positive integer $n$, show that a group of order
$n$ has exactly $2^{n-1}$ {\em subsets} which contain $1_G$.  Hint: Use
induction.
\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 if $G$ has order $p^n$ (for $n \geq 1$) then $G$ has an element
  whose order is $p$.
\end{problem}

\begin{problem} Let $G$ be a finite group and $H \subset G$ be an
nonempty subset.
  Suppose that $H$ is closed under multiplication. % and contains $1_G$.
  Show that $H$ is in fact a subgroup of $G$.
\end{problem}


%\begin{problem}
%Let $G$ be a group.  We define the {\em
%center} of $G$ to be
%$$
%Z(G) = \{x \in G: xg = gx \text{ for all } g \in G\}.
%$$
%Prove that $Z(G)$ is a subgroup of $G$.
%\end{problem}

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


\end{document}