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{\bf
\noindent
ALGEBRA 250A (Prof. De Loera) \hskip 4cm \hfill \null \\
Graduate Proficiency Exam  \hskip 5.4cm Fall 2005 \hfill \null \\}

\noindent
These problems are intended to help you review material
covered in the prelim. Some problems are harder and longer than what
you will find in the exam. Try to write the solutions as you would
in the exam: Write all details when solving every problem. 
Be organized and use the notation appropiately. Initially try to solve 
the problems without any assistance. These problems cover only the
first half of MAT 250A. The rest will be covered later in September.

\begin{itemize}

\item {\bf Group Theory}

\begin{enumerate}

\item Prove: A subgroup of a cyclic group is cyclic.

\item Let G be a cyclic group of order 20. Indicate each element that
generates $G$. List the distinct subgroups of $G$ and a diagram of
inclusion among them.


\item Let $\sigma$ be the permutation of $S_9$ given by  
$ \left[ \begin {array}{ccccccccc} 1&2&3&4&5&6&7&8&9
\\\noalign{\medskip}9&1&5&8&6&3&2&4&7\end {array} \right]$

Determine the order of $\sigma$. Is it an even or odd permutation?

\item Prove or disprove:

a) A finite group always has a non-trivial
  center.

b) A finite group always has a subgroup of order $k$ for each divisor
$k$ of the order of $G$.

c) All groups of order 325 must be Abelian.

d) A group of order 12 must contain a normal subgroup of order 2 or 4.


\item Let $G$ be a finite $p$-group. Show that

a) If $|G|=p$, then $G$ is cyclic.

b) Show that if $G$ is non-trivial, then the center of $G$ is
non-trivial.

c) Show that if $|G|=p^2$, then $G$ is Abelian.

d) Every finite p-group is solvable.

\item  Show that $Z_m \times Z_n$ is isomorphic to $Z_{nm}$ if and
  only if $gcd(n,m)=1$.

\item Show that $Aut(Z_n)$ is always Abelian, but in general not cyclic.
Show that if $p$ is a prime, then $Aut(Z_p)$ is cyclic of order $p-1$.

\item For a group $G$ denote by $G'$ the commutator subgroup. Show
  that $G'$ is a normal subgroup of $G$. Show that $G/G'$ is Abelian.

\item Let $n\geq 3$ and $G=S_n$, Show that $G'=A_n$.

\item Let $G$ be a finite group. Let $K,S$ be subgroups of $G$ and
suppose that $K$ is contained in the normalizer of $S$ in $G$. Show
that $KS$ is a subgroup of $G$.

\item Show that if $|G|=24$, then $G$ is not simple; that is, $G$
contains a non-trivial proper normal subgroup.

\item Show that if $|G|=100$, then $G$ is not simple. 

\item Let $G$ be the group defined by generators $a,b,c$ and relations

$$ a^2=b^2=c^2=(ab)^3=(ac)^2=(bc)^3=id.$$

a) Determine the order and the index of the subgroup $H=<a,b>$ of $G$.

b) Show that $G$ is isomorphic to the symmetric group $S_4$.

\item If $p>q$ are prime numbers, a group of order $pq$ has at most
one subgroup of order $p$.

\item Find subgroups $H,K$ of the dihedral group $D_4$ such that
they are $H$ is normal in $K$, $K$ is normal in $D_4$ but $H$ is not
normal in $D_4$.

\item True of False? Give either a proof or a counterexample:

a) Every finite Abelian p-group is generated by its elements of
maximal order

b) Every short exact sequence of Abelian groups is split.

c) All solvable groups are nilpotent.

d) The group $A_4$ has no subgroup of order 6.

e) If $H$ is a normal subgroup of a finite group $G$, then $G$ has a
subgroup that is isomorphic to $G/H$.

\item Determine all irreducible complex representations of the
  dihedral group $D_n$.

\item Let $B$ the subgroup of $GL(C,n)$ consisting of all upper
  triangular matrices. Show that $B$ is a solvable group

\item Prove that every complex representation $T$ of an Abelian group
  $G$ has a one-dimensional $T$-invariant subspace.

\item Determine up to isomorphism all Abelian groups of order 64.

\item If $N$ is a normal subgroup of order $p$ (prime) of a group $G$
  of order $p^n$, then $N$ is in the center of $G$.

\item How many elements of order seven are there in a simple group of
order 168?

\item Find the Sylow 2-subgroups and Sylow 3-subgroups of $S_3,S_4$

\item Classify up to isomorphism all groups of order 30.

\item Any group of order $p^2q$ is solvable.

\item Let $G$ be the group of $2 \times 2$ non-singular matrices with
entries in a field of 2 elements (operation is matrix multiplication). 
What is the order of $G$? What are the possible orders of subgroups of
$G$? 


\item If in the group $G$, $a^5=1$ and $aba^{-1}=b^2$ and $b$ is not
the identity, find the order of $b$.

\item Find the number of elements in the conjugacy class of
  $g=(123)(456)$ in $S_6$. Find the order of the centralizer of $g$.

\item Find the order of a Sylow $5$-subgroup of $GL_2(Z_5)$. Find one
such subgroup explicitly.

\item Let $M$ be an Abelian group with generators $a,b,c,d,e$
  satisfying the relations:

$a-7b+14d-21e=0$, $5a-7b-2c+10d-15e=0$, $3a-3b-2c+6d-9e=0$,
  $a-b+2d-3e=0$.

Give the structure of $M$ as a direct sum of cyclic groups. Give the
invariant factors and the elementary divisors of $M_t$, the torsion
subgroup of $M$. What is the rank of the torsion free part of $M$?

\end{enumerate}


\item{\bf Ring Theory}

\begin{enumerate}

\item Show that every ideal in the integers is principal.

\item Consider $Z_{85}$ as a ring. Which elements have multiplicative
inverses? 24 is one of them, find its inverse.

\item Let $R$ be the ring of Gaussian integers, i.e.

$R=\{a+bi | a,b \in {\mathbb Z}, i^2=-1\}$

a) Show that the principal ideal (2+i) is a maximal ideal of $R$.

b) What is the quotient ring $R/(2+i)$?

\item Show that $f(x)=3x^5-2x^3+4x^2-8x+6$ and $g(x)=3x^3+4x^2-5x+1$
are irreducible in ${\mathbb Q}[x]$.

\item Prove that in a commutative ring with identity every maximal
ideal $M$ is a prime ideal. 


\item Consider the homomorphism of polynomial rings

$$ \phi:\ {\mathbb Q}[x,y,z] \rightarrow {\mathbb Q}[t], \ x \rightarrow t^2, y
  \rightarrow t^3, z \rightarrow t^5$$

(where $\phi(q)=q$ for all rational numbers), let $I$ be the kernel of
  $\phi$.

Show that $I$ is a prime ideal, but it is neither maximal nor
principal.


\item Let $I$ be an ideal generated by square free monomials inside
${\mathbb Q}[x_1,x_2,\dots,x_n]$. Show that
$I$ can be written as a finite intersection of ideals of the form
  $<x_{i_1},\dots, x_{i_d}>$.

\item Prove or disprove: 

a) Every subring of ${\mathbb Q}[x_1,x_2]$ is finitely generated.

b) Every subring of ${\mathbb Q}[x_1,x_2]$ is a unique factorization
domain.

c) Every non-zero homomorphic image of a local ring is local.


d) ${\mathbb Q}[x]$ is a PID and a UFD

e) ${\mathbb Z}[x]$ is a PID and a UFD

f) Let $I$ be an ideal of $R$, a commutative ring with identity,
then $R/I$ is a field if and only if $I$ is a maximal ideal.

g) f irreducible in ${\mathbb Z}[x]$ implies that ${\mathbb Z}[x]/I$
is a field.

\item Find all two-sided ideals of the ring of $2\times 2$ matrices
over $Z_5$.

\item Prove that there does not exist any ideal $I$ such that 
$(x^4+x+1) \subset I \subset K[x]$ where $K$ is the field with 2
  elements and the inclusions are proper.

\end{enumerate}

\end{itemize}

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