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\begin{document}

\maketitle

\begin{center}
{{\Large\bf MODERN ALGEBRA II}\\
\medskip

{\Large\bf HOMEWORK ASSIGNMENT VII}\\
\medskip
{\bf (Turn in underlined problems.)}}
\end{center}

\vspace{.25in}

\noindent
{\bf 
\begin{tabbing}
\underline{\large\bf CHAPTER}\makebox[2in]{}\=\underline{\large\bf PROBLEMS}\\

\bigskip
\noindent
2 \> \underline{2.74}
\end{tabbing}

}
\vspace{.25in}

\noindent
\underline{\large\bf ADDITIONAL PROBLEMS}
\vspace{.25in}

{\bf 

\underline{1.}
Find all abelian groups of order $144$
(up to isomorphism).

\medskip

\underline{2.} 
Let ${\BF}$ be a field. Show that any finite subgroup
of the multiplicative group ${\BF}-\{0\}$
%$\langle F-\{0\};\cdot, {}^{-1},1\rangle$ 
is cyclic.
(Hint: Let $H$ be a finite subgroup of ${\BF}-\{0\}$.
$H$ is abelian, so it can be represented
as a direct product of cyclic groups
${\BZ}_{d_1}\times {\BZ}_{d_2}\times 
\cdots\times {\BZ}_{d_n}$ with $d_1 \mid 
d_2 \mid \cdots \mid d_n$.
Show that the $d_1d_2\cdots d_n$
elements of $H$ are distinct 
roots of $x^{d_n} - 1 = 0$.
Conclude that $d_1d_2\cdots d_n \leq d_n$.
Conclude that $H\cong {\BZ}_{d_n}$.)

\medskip

\underline{3.} 
An element $r\in {\BZ}_n$ is
called a {\it primitive root modulo $n$}
if the set $\{1, r, r^2, \ldots, r^{n-2}\}$
equals the set $\{1, 2, \ldots, n-1\}$.
(For example, $2$ is a primitive root
modulo $5$ since 
$\{1, 2, 2^2, 2^3\} = \{1, 2, 4, 3\}$.)
Prove that if $p$ is prime, then there
is a primitive root modulo $p$.
Find a primitive root modulo $23$.

\medskip

\underline{4.} 
Show that if $G, H$ and $K$ are finite abelian groups
and $G\times H \cong G\times K$, then 
$H\cong K$.
Show that this result may fail if $G$ is an 
infinite abelian group.
(Hint for the second statement: take
$G = \prod_{i=1}^{\infty} {\BZ}_2 = {\BZ}_2\times
{\BZ}_2\times\cdots$.)

\medskip

5. Show that $\langle {\BQ}; +, -, 0\rangle$
is indecomposable.

\medskip

6. Show that $\langle {\BQ}/{\BZ}; +, -, 0\rangle$ is decomposable.

\medskip

7. Determine whether or not $\langle{\BR}; +, -, 0\rangle$
is decomposable.

\medskip

For Graduate Credit: (Problem G only)

G. Let $H$ is a finite group, and
let $n_k$ be
the number of elements of order $k$ in $H$.
Define $o_H = \langle n_1, n_2, \ldots\rangle$ 
to be the sequence
that lists the number of elements of each order in $H$.
(Example: if $H = {\BZ}_6$, then
$o_H = \langle 1, 1, 2, 0, 0, 2, 0, 0, \ldots\rangle$.)
Show that if $H$ and $K$ are finite abelian 
groups and $o_H = o_K$, then $H\cong K$.

\medskip

8. Show that there are groups $H$ and $K$
of order $27$ such that 
$o_H = o_K$ but $H\not\cong K$.
(This does not contradict Problem G,
since one of these groups is abelian and one is not.)

\end{document}

