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\begin{center} {\Large\bf Honors Algebra I \\ Assignment 8} \\ Due,
October 30 \end{center}
\begin{quote}
She guessed at it all, what might live, moving purposefully or
drifting aimlessly, under the deep water around her, but she
didn't think too much about any of it. It was enough to be
aware of the million permutations possible around her, and
take comfort in knowing she would not, and really could not,
know much at all.
\\ Dave Eggers, The Circle
\end{quote}
\ben
\item Ima Geek is given a group $G$ with $125=5^3$ elements
and is trying to follow one of the cases of the proof given in class
to get $G$ isomorphic to $Z_{25}\times Z_5$. Ms. Geek doesn't
realize that the group actually {\em is} $Z_{25}\times Z_5$
and she picks $a=(3,4)$ as her first element. Show that $a$
has order $25$. Set $N=\{0,a,\ldots,24a\}$. Show that every
element of $G/N$ can be written as $\ol{(0,k)}$ for a
unique $0\leq k<5$. Now she picks $b=(11,1)$ as her second
element. Find $0\leq k<5$ so that $\ol{b}=\ol{(0,k)}$. Follow
the argument given in class to find an explicit $c\in G$ with
$c=b-ia$ for an explicit $i$ so that $o(c)=5$ in $G$.
Find $0\leq i < 25$ and $0\leq j < 5$ with $ai+cj = (1,0)$ in
$Z_{25}\times Z_5$.
\item Let $G,H$ be groups under multiplication with identities
$e_G,e_H$ respectively.
Let $J=G\times H$
be their direct product. Set
\[ \Lambda = \{ (g,e_H)\in J: g\in G \} \]
\ben
\item Prove $\Lambda$ is a subgroup of $J$.
\item Prove $\Lambda$ is a normal subgroup of $J$.
\item Let $n\geq 5$.
Prove (using the above and results proven in class) that
$S_n$ is {\em not} isomorphic to $(S_n/A_n)\times A_n$.
\een
\item Three problems about manipulating products of cyclic groups.
\ben
\item
Write $Z_2\times Z_2\times Z_2\times Z_9\times Z_5 \times Z_{25}$
as the product of cyclic groups $Z_{a_i}$, $1\leq i\leq s$ (you find the
$s$) with $a_i$ dividing $a_{i+1}$ for all $1\leq i < s$.
\item Write $Z_{4}\times Z_{40}\times Z_{200}\times Z_{1400}$ as the
product of cyclic groups of prime power order. (Prime power includes
primes themselves.)
\item Write $Z_5\times Z_6\times Z_7$ in {\em both} of the above forms.
\een
\een
\begin{quote}
Nothing is more fruitful - all mathematicians know it - than those
obscure analogies, those disturbing reflections of one theory in
another; those furtive caresses, those inexplicable discords; nothing
also gives more pleasure to the researcher. The day comes when the
illusion dissolves; the yoked theories reveal their common source
before disappearing. As the {\em Gita} teaches, one achieves
knowledge and indifference \footnote{Some translators of the
original Sanskrit use ``detachment" here.} at the same time.
\\ Andr\'e Weil
\end{quote}
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