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\begin{center} {\Large\bf Algebra , Assignment 6} \\ Due,
Friday, Oct 9 \end{center}
\begin{quote}
She had been born with a map of time in her mind. She
pictured other abstractions as well, numbers and the
letters of the alphabet, both in English and in Bengali.
Numbers and letters were like links on a chain. Months
were arrayed as if along an orbit in space.
\\ Jhumpa Lahiri, The Lowland
\end{quote}
\ben
\item Let $\sig: G\ra G$ be an automorphism of $G$. Let
$x,y\in G$ with $y=\sig(x)$.
\ben
\item Assume $x^s=e$. Prove $y^s=e$.
\item Assume $x^s\neq e$. Prove $y^s\neq e$.
\item Show $o(x)=o(y)$ (You can assume they are both finite.)
\een
\item Let $\sig\in S_n$ be given by $\sig(i)=n+1-i$ (so $1\cdots n$
is ``reversed.") When $n=403$ is $\sig$ even or odd? For which
$n$ is $\sig$ even?
\item Here we write perutations in terms of their disjoint cycles, with commas
eliminated.
\ben
\item Given $\sig=(12)(34)$ and $\gam=(56)(13)$ find $\tau\in S_6$ with
$\tau^{-1}\sig\tau=\gam$
\item Argue that with $\sig=(123)$ and $\gam=(13)(578)$ there is no $\tau$ with
$\tau^{-1}\sig\tau=\gam$
\item Set $\sig=(12)(34)$. In $S_8$ precisely how many $\gam$ have the property
that
$\tau^{-1}\sig\tau=\gam$ for some $\tau$?
\een
\item Consider the symmetries of the square. Use the table and
notation from the solution to Assignment 1 on the website.
For each symmetry $v$ find the
conjugacy class $C(v)$ and the Normalizer $N(v)$ as defined
in \S 2.11. Check by calculation that the class equation holds,
that is, that
\[ |G| = \sum \frac{|G|}{|N(v)|} \]
where the sum ranges over one element $v$ in each conjugate class.
\item In $S_4$ set $\sig=(12)(34)$, $\tau=(13)(24)$, $\gam=(14)(23)$, $e$ the
indentity. Let $H=\{e,\sig,\tau,\gam\}$. Give the multiplication table for
$H$. Show that $H$ is a subgroup of $S_4$. Show that $H$ is a subgroup of
$A_4$. Show that $H$ is a normal subgroup of $A_4$. Show that
$H$ is a normal subgroup of $S_4$. Set $G_1=(Z_4,+)$ and
$G_2=(Z_2\times Z_2,+)$. (We call $G_2$ the viergr\"uppe.) Which of
$G_1,G_2$ is $H$ isomorphic to? Given an explicit (there are many
correct answers) isomorphism $\phi$ from $H$ to whichever ($G_1$ or $G_2$) it is
isomorphic to.
\item Let $GL_n(R)$ be, as usual, the $n\times n$ nonsingular
matrices over the Reals, under multiplication. Suppose
$A\in GL_n(R)$ has real eigenvectors $\vec{v_1},\ldots,\vec{v_n}$
with distinct real eigenvalues $\lam_1,\ldots,\lam_n$, so that
$A\vec{v_i}=\lam_i\vec{v_i}$ for $1\leq i\leq n$. Recall from
Linear Algebra that this implies that the vectors $\vec{v_i}$
form a basis for $R^n$. Let $\sim$
be the conjugacy relation, that is, $B\sim A$ means $B=P^{-1}AP$ for
some matrix $P$.
\ben
\item Suppose $B\sim A$. Prove that $B$
has the same eigenvalues $\lam_1,\ldots,\lam_n$. That is, show
that there are real eigenvectors $\vec{w_1}\ldots,\vec{w_n}$ with
$B\vec{w_i}=\lam_i\vec{w_i}$ for $1\leq i\leq n$.
\item (*) Conversely, Suppose $B$ has real eigenvectors $\vec{w_1},\ldots,\vec{w_n}$
with distinct real eigenvalues $\lam_1,\ldots,\lam_n$, the same values
as for $A$. Prove that $B\sim A$.
\een
\een
\begin{quote}
Bach, Mozart, Schubert - they will never fail you. When you perform
their work properly it will have the character of the inevitable, as
in great mathematics, which seems always to be made of pre-existing
truths. \\ E. L. Doctorow, {\em City of God}
\end{quote}
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