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\begin{center} {\Large\bf Honors Algebra \\ Assignment 2} \\ Due,
Friday, Sept 11 in Recitation \end{center}
\begin{quote}
Sandwiched as we are between the ``everything" that is behind us and the
``zero" beyond us, ours is an ephemeral existence in which there is
neither coincidence nor possibility.
\\ Haruki Murakami, A Wild Sheep Chase
\end{quote}
\ben
\item In $S_3$ (reminder, this is our standard notation for
the permutations on $\{1,2,3\}$) show that there are four
elements $x$ satisfying $x^2=e$ and three elements $x$ satisfying
$x^3=e$.
\item In $Z_{13}^*$ let $H=\{1,5,12,8\}$. List the right cosets
$Ha$.
\item Let $G$ be the symmetries of the square. (See the solutions
to assignment 1 for a table.) Let $H=\{I,V\}$. List the right
cosets $Ha$ and the left cosets $aH$. Do the same with $H=\{I,R,S,T\}$.
\item Let $G=S_n$ and let $H=\{\sig\in G: \sig(1)=1\}$. Let $\tau\in G$
and suppose $\tau(i)=1$. By $\tau H \tau^{-1}$ we mean all elements
of the form $\tau\sig\tau^{-1}$ where $\sig\in H$.
\ben
\item Show that if $\gam\in \tau H\tau^{-1}$ then $\gam(i)=i$
\item Show that if $\gam(i)=i$ then $\gam\in \tau H\tau^{-1}$
\een
\pagebreak
\item Give the elements and the multiplication table for $Z_{15}^*$.
(The elements of $Z_n^*$ are those $i$, $1\leq i \leq n-1$ which are
relatively prime\footnote{Positive integers $m,n$ are called relatively
prime if they have no common factor. Do not confuse this with primality!}
to $n$. Multiplication is modulo $n$.)
Find the order $o(a)$ of each element $a$. (The order of $a$ is the
least positive integer $n$ so that $a^n=1$.)
\item The {\em center} $Z$ of a group $G$ is the set of all $z\in G$
with the property that $zg=gz$ for all $g\in G$. Prove that $Z$ is
a subgroup of $G$. Prove the $Z$ is Abelian.
\een
\begin{quote}
I cannot pretend I am without fear. But my predominant feeling is one of
gratitude. I have loved and been loved; I have been given much and I have
given something in return; I have read and traveled and thought and written.
I have had an intercourse with the world, the special intercourse of writers
and readers.
\par Above all, I have been a sentient being, a thinking animal, on this
beautiful planet, and that in itself has been an enormous privilege and
adventure.
\\ Oliver Sacks, 1933-2015
\end{quote}
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