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\title{Math 120: Homework 6}
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\maketitle

\begin{itemize}[noitemsep]
  \item Section 4.3 \# 28,34
  \item Section 4.4 \# 2,13
  \item Section 4.5 \# 13,25
%  \item Section 7.4 \# 37
  \item Section 7.5 \# 3
\end{itemize}

\bigbreak\noindent
{\textbf{Problem 4.3 \#28.}} Let $p$ and $q$ be distinct primes with $p <
q$. Prove that a nonabelian group $G$ of order $p q$ has a nonnormal subgroup
of index $q$, so there exists an injective homomorphism $G \longrightarrow
S_q$. Deduce that $G$ is isomorphic to a subgroup of the normalizer in $S_q$
of the cyclic group generated by the $q$-cycle $(1, 2, \cdots, q)$.

\bigbreak\noindent
{\textbf{Problem 4.3 \# 34.}} Prove that if $p$ is a prime and $P$ is a
subgroup of $S_p$ of order $p$ then $| N_{S_p} (P) | = p (p - 1)$. [Argue that
every conjugate of $P$ contains exactly $p - 1$ $p$-cycles and use the formula
for the number of $p$-cycles to compute the index of $N_{S_p} (P)$ in $S_p$.]

\bigbreak\noindent
{\textbf{Problem 4.4 \# 2}}. Prove that if $G$ is an abelian group of order
$p q$, where $p$ and $q$ are distinct primes then $G$ is cyclic.

\bigbreak\noindent
{\textbf{Problem 4.4 \# 13.}} Let $G$ be a group of order $203$. Prove that
if $G$ has a normal subgroup $H$ of order $7$ then $H \subseteq Z (G)$.

\bigbreak\noindent
{\textbf{Problem 4.5 \# 13.}} Prove that a group of order 56 has a normal
$p$-Sylow subgroup for some prime $p$ dividing $56$.

\bigbreak\noindent
{\textbf{Problem 4.5 \# 25.}} Prove that if $G$ is a group of order 385
then $Z (G)$ contains a 7-Sylow subgroup and an 11-Sylow subgroup is normal in
$G$.

% \bigbreak\noindent
% {\textbf{Problem 7.4 \#37.}} A commutative ring $R$ is called a
% {\textit{local ring}} if it has a unique maximal ideal. Prove that if $R$ is a
% local ring with maximal ideal $M$ then every element of $R - M$ is a unit.
% Prove conversely that if $R$ is a commutative ring with unit such that the
% nonunits of $R$ form an ideal, then $R$ is a local ring with a unique maximal
% ideal $M$.

\bigbreak\noindent
{\textbf{Problem 7.5 \# 3.}} Let $F$ be a field. Prove that $F$ contains a
unique smallest subfield $F_0$ and that $F_0$ is isomorphic to either
$\mathbb{Q}$ or $\mathbb{Z}/ p\mathbb{Z}$ for some prime $p$.

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