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Fall 1993
\begin{center} MA Comprehensive Exam in Algebra\\
\end{center}
Directions: Do all the problems. Write clearly and concisely. Be
clear in your writing about what results you are using.\
The examiners have made every effort to make sure that the
problems are correct. However, if you are convinced that a
problem contains a misprint, explain why you think so, formulate
a correct and non-trivial version of the problem, and do that
instead.\\
$Q$ = the rational numbers\
$\bf{R}$ = the real numbers\
$C$ = the complex numbers
\begin{enumerate}
\item
(a) If $G$ is a group which contains only a finite number of
subgroups, show that $G$ is finite.\\
(b) Describe all groups which contain no proper subgroups.\\
(c) Describe all groups $G$ which contain exactly one proper
non-trivial subgroup.
\item
(a) Show that the alternating group $A_4$ contains a normal
subgroup of order 4.\\
(b) Prove or disprove: There is a homomorphism from $A_4$ onto
$Z/(3)$ (where $Z/(3)$ is the group of integers mod 3).\\
(c) Prove or disprove: There is a homomorphism from $A_4$ onto
$Z/(4)$.\\
(d) Prove or disprove: There is a homomorphism from $A_4$ onto
$S_3$, the symmetric group on three letters.
\item
(a) If $R$ is an integral domain and $I$ and $J$ are non-zero
ideals of $R$, prove that $I \cap J \ne 0$.\\
(b) If $R = Q[x]$, find all proper ideals $I$ and $J$ such that
$I \cap J = (x^2 - x - 1)$ (the principal ideal generated by $x^2
- x - 1$). Is your answer the same if $R = \bf{R}$$[x]$?\\
(c) If $R=Q[x]$, prove or disprove the statement that $R = (x^4 -
x^2) + I$ for some proper ideal $I$.
\item
Prove that a finite integral domain is a field.
\item
Let $V$ be a finite dimensional vector space over $C$ and $T: V
\rightarrow V$ be a linear transformation satisfying the equation
$T^4 = I$.\
(a) Prove that $T$ can be represented by a diagonal
matrix.\
(b) Give an example to show that if $V$ is a finite dimensional
vector space over $\bf{R}$, and $T$ is as above, then $T$ need not
be diagonalizable.
\item
Suppose $f:V \rightarrow W$ and $g:W \rightarrow X$ are linear
transformations, where $V$, $W$ and $X$ are vector spaces of
dimensions $n$, $k$ and $m$ respectively. Suppose that $f$ and
$g$ have ranks $r_f$ and $r_g$ respectively. What are the
possible ranks for the composite $g f$? Justify your answer.
\end{enumerate}
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