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\begin{center} {\Large\bf Honors Algebra I\\ Assignment 7} \\ Due,
Friday, October 23 \end{center}
\begin{quote}
Comstock grins and says, `You sound awfully sure of yourself, Waterhouse!
I wonder if you can get me to feel that same level of confidence.'
\\ Waterhouse frowns at the coffee mug. `Well, it's all in the math,' he says,
`If the math works, why then you {\em should} be sure of yourself. That's
the whole point of math.'
\\ from {\em Cryptonomicon} by Neal Stephenson
\end{quote}
\ben
\item Let $G$ denote the set of linear functions $f(x)=mx+b$ on
the real line with $m\neq 0$. Denote such a function by $(m,b)$.
Define a product $f^*g$ as the function $h(x)=g(f(x))$. (Check
assignment 1 and the solutions for earlier work on this group.)
Recall $C(f),N(f)$ denote the conjugate class and
the normalizer of $f$.
\ben
\item Describe $C(f)$ and $N(f)$ when $m\neq 1$ and $b\neq 0$.
\item Describe $C(f)$ and $N(f)$ when $m\neq 1$ and $b=0$.
\item Describe $C(f)$ and $N(f)$ when $m=1$ and $b\neq 0$.
\item Describe $C(f)$ and $N(f)$ when $m=1$ and $b=0$.
\item Describe $Z[G]$, the center of the group.
\een
\item Let $\sig\in S_n$ be (in cycle notation)
\[ \sig = (12\cdots n) \]
\ben
\item Describe in words the $\gamma\in C(\sig)$.
\item Find $|C(\sig)|$.
\item Deduce $|N(\sig)|$.
\item (*) Describe $N(\sig)$ explicitly. (Idea: Since you already
know that $N(\sig)$ has precisely woggle elements, you should look
for woggle distinct elements that are in $N(\sig)$ and then you have
them all.)
\een
\item Let $o(G)=p^n$, $p$ prime. Prove that $o(Z[G])\neq p^{n-1}$.
(Idea: Examine the proof that groups of order $p^2$ are Abelian.)
\item Let $G=Z_5\times Z_{25}\times Z_{125}$.
Find the order of $(3,10,12)$. Give a good description and a precise
count on those $(a,b,c)\in G$ order $25$.
\item Let $H$ be a normal subgroup of $G$ (all under multiplication).
Let $g\in G$ and let (as usual) $\ol{g}$ denote the corresponding
element in $G/H$. Suppose $\ol{g}$ has order $a$ (in $G/H$) and
$g^a$ has order $b$ (in $G$). Prove $g$ has order $ab$ in $G$.
\item (Warning: Lots of grunt work here.) Let $H$ be the subgroup
of $S_4$ which is the normalizer of $(12)(34)$, as discussed in class.
Is this group isomorphic to the group of symmetries of the square, as
given in the solutions to Assignment 1. If they are not isomorphic, prove it. If they
are isomorphic, give an explicit isomorphism.
\een
\begin{quote}
The universe is not only queerer than we suppose but queerer than we {\em can}
suppose. \\ -- J.B.S. Haldane
\end{quote}
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