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\centerline{ \huge\bf {Algebra Preliminary Exam}  }
\vspace{5mm}
\centerline{ \Large\bf {August, 2003} } 

\vspace{5mm}
 \ben
\item[1)]   Let $GL_n(F_p)$ denote the group of invertible $n \times n$ matrices
with entries in the field of $p$ elements $F_p = Z/pZ$.  Equivalently,
$GL_n(F_p)$ denotes the group under composition of the 1-1 and onto linear
transformations from $F_p^n$ to $F_p^n$.
\ben
\item[a)]  Determine the order of $GL_3(F_p)$. 
\begin{comment}
 [pick a basis e_1, e_2, e_3 of
F_p^3, map e_1 to any element not zero, then map e_2 to any element not in
the subspace spanned by the image of e_1, etc.  Have (p^3 -1)(p^3 -p)(p^3 -
p^2) choices. ]
\end{comment}
\item[b)]  Show that the subgroup $U$ of $GL_3(F_p)$ consisting of matrices of the
form $I + N$ where $N$ is upper triangular with zero diagonal, is a
$p$-Sylow subgroup of $GL_n(F_p)$. 
\item[c)] Show that $U$ is nilpotent but not abelian.
\begin{comment}
 [the subgroup has order p^3, the power
of p dividing the cardinality of GL_3(F_p)]
\end{comment}
\een
\item[2)]  If $G$ is a  group, let $Aut(G)$ be the group of automorphisms of
$G$.  Define $C: G  \to Aut(G)$ to be the function defined by $C(g)(x) = gxg^{-1}$
for $g, x$ in $G$.
\ben
\item[a)]  Show that $C$ is a homomorphism.

\item[b)]  Describe the kernel of $C$. 
\begin{comment}
 [the center of G]
\end{comment}
\item[c)]  Show that if $G = S_3$, the symmetric group on three symbols, then $C$
is an isomorphism. 
\begin{comment}
[by looking at generators of G, there are
at most 6 automorphisms, and C is 1-1]
\end{comment}
\een

\item[3)]  Define the semi-direct product of two groups as follows:  let
$(G, *)$ and $(A, \cdot)$ be groups, and $\alpha: A \to Aut(G)$ a
homomorphism.  Then
\[G \rtimes_{\alpha} A = \{ (g, a) \in G \times A \} \]
with multiplication:
\[ (g_1, a_1)\cdot (g_2, a_2) = (g_1 * \alpha(a_1)(g_2), a_1 \cdot a_2 ) .\]
\ben
\item[a)]  Show that if $\alpha$ is trivial, then $G \rtimes_{\alpha} A\cong G \times
A$. 
\item[b)]  Find six pairwise non-isomorphic groups of order 42, all of which are
semi-direct products.
\begin{comment}
 [C_7 \rtimes C_6, (C_7 \rtimes C_2)\times C_3 = D_7
\times C_3, (C_7 \rtimes C_3) \times C_2, C_{42}, C_7 \times D_3, D_21 ]
\end{comment}
\een
\item[4)]  If $R$ is a commutative ring with 1 and $I, J$ are ideals of $R$ so that
$I + J = R$, then
\[ R/(I \cap J) \cong R/I \oplus R/J .\]
Show that when $R = Z$, the integers, this theorem is equivalent to a
well-known theorem in elementary number theory.  Explain.
\begin{comment}
 [chinese remainder
theorem]
\end{comment}
\item[5)]  Find the Galois group over $F_2$, the field of 2 elements, of the
irreducible polynomial $f(x) = x^5 + x^2 + 1$.
\begin{comment}
 [cyclic of order 5, generated
by a  maps to a^2]
\end{comment} 
\item[6)] How many similarity classes of matrices with complex entries have characteristic polynomial $(x-1)(x-2)^2(x-3)^3$? Explain.

\item[7)] Let $F$ be a finite field.
\ben
\item[a)] Define the characteristic of $F$.
\item[b)] Prove that the characteristic of $F$ must be a prime $p$.
\item[c)] Prove that the cardinality of $F$ must be a power of $p$.
\item[d)] Show that $F$ is the splitting field of some irreducible polynomial.
\een
\item[8)] Suppose $R$ is a ring.
\ben
\item[a)] What does it mean to say that $R$ is a {\it Euclidean Ring\/}? 
\item[c)] What does it mean to say that $R$ is a {\it Principle Ideal Domain\/}? 
\item[c)] Prove that every Euclidean Ring is a Principle Ideal Domain.
\een
\een
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