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\title{Math 120: Homework 8}
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\maketitle

\begin{itemize}[noitemsep]
  \item Section 5.4 \#5
  \item Section 5.5 \#8,11
  \item Section 8.2 \#1,4
  \item Section 8.3 \#2
\end{itemize}

\bigbreak\noindent
{\textbf{Problem 5.4 \#5.}} Prove that $A_n$ is the commutator subgroup of the symmetric group $S_n$ 
for $n\geqslant 5$.

\bigbreak\noindent
{\textbf{Problem 5.5 \#8.}} Construct a nonabelian group of order 75.
Classify all groups of order 75 (there are 3 of them).

\bigbreak\noindent
{\textbf{Problem 5.5 \#11.}} Classify groups of order 28 (there are four
isomorphism types).

\bigbreak\noindent
{\textbf{Problem 8.2 \#1.}} Prove that in a principal ideal domain $R$ two
ideals $(a)$ and $(b)$ are comaximal if and only if the greatest common
divisor of $a$ and $b$ is~$1$.

\bigbreak\noindent
{\textbf{Problem 8.2 \#4.}} Let $R$ be an integral domain. Prove that if
the following two conditions hold then $R$ is a principal ideal domain.

(i) any two nonzero elements $a$ and $b$ have a greatest common divisor which
can be written in the form $r a + s b$ for some $r, s \in R$, and

(ii) If $a_1, a_2, \cdots$ are nonzero elements of $R$ such that $a_{i + 1}
|a_i$ for all $i$, then there is a positive integer $N$ such that $a_n$ is a
unit times $a_N$ for all $n \geqslant N$.

\bigbreak\noindent
{\textbf{Problem 8.3 \#2.}} Let $a$ and $b$ be nonzero elements of the
unique factorization domain $R$. Prove that $a$ and $b$ have a least common
multiple and describe it in terms of the prime factorizations of $a$ and $b$.

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