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\centerline{Mathematics 243, section 3 -- Algebraic Structures}
\centerline{Problem Set 8}
\centerline{{\bf due:}  Friday, November 16}
\vskip 10pt
\noindent
{\it `A' Section}
\bigskip
\begin{enumerate}
\item[1.]  Consider the $2\times 2$ matrices $I_2,S,X,Y,D,T$ defined by 
\begin{align*} 
I_2 = \begin{pmatrix} 1&0 \\ 0&1\end{pmatrix}, &\quad S = \begin{pmatrix} 0&-1 \\ 1&0\end{pmatrix}\\
X = \begin{pmatrix} 1&0 \\ 0&-1\end{pmatrix}, &\quad Y = \begin{pmatrix} -1&0 \\ 0&1\end{pmatrix}\\
D = \begin{pmatrix} 0&1 \\ 1&0\end{pmatrix}, &\quad T = \begin{pmatrix} 0&-1 \\ -1&0\end{pmatrix}
\end{align*}
Let $G = \{I_2,S,S^2,S^3,X,Y,D,T\}$.  Construct the operation table for matrix multiplication
on this set and verify that $G$ is a group under this operation.
\item[2.]   
\begin{enumerate}
\item[a.] Let $G = \mathbb{Z}_{12}$ under the operation of \emph{addition} modulo 12.  Determine
the cyclic subgroups $\langle [a]\rangle$ for all $[a] \in \mathbb{Z}_{12}$.    
\item[b.] Conjecture a general formula for the number of elements in $\langle [a]\rangle$ in 
terms of the integers $a$ and $12$.  Check out your conjecture on the corresponding list of 
cyclic subgroups of $\mathbb{Z}_{20}$ constructed in class on November 14.  
\end{enumerate}
\end{enumerate}
\vskip 10pt
\noindent
{\it `B' Section}
\vskip 10pt
\begin{enumerate}
\item[1.]  Let 
$$G = \left\{\begin{pmatrix} 0&a&b\\ 0&0&c\\ 0&0&0\end{pmatrix} \mid a,b,c \in \mathbb{R}\right\}$$
Show that $G$ is a group under the operation of matrix addition.
\item[2.]  Let 
$$G = \left\{\begin{pmatrix} 1&a&b\\ 0&1&c\\ 0&0&1\end{pmatrix} \mid a,b,c \in \mathbb{R}\right\}$$
Is $G$ is a group under the operation of matrix multiplication.  If so, say why; if not say which 
properties in the definition fail.  
\item[3.]  Let $G$ be a group, and consider the relation $R$ on $G$ defined by 
$xRy \leftrightarrow y = axa^{-1}$ for some $a\in G$.  (If $xRy$ is true, the $a$ that works
in the equation will depend on $x,y$.)   $R$ is called the \emph{conjugacy} relation on $G$.
\begin{enumerate}
\item[a.] Show that conjugacy is an equivalence relation on $G$.
\item[b.] Show that $G$ is an abelian group if and only if the equivalence classes
for the conjugacy relation satisfy $[x] = \{x\}$ for all $x \in G$.
\item[c.]  More generally, show that if $[x] = \{x\}$ for some element $x \in G$, then 
$xa = ax$ for all $a \in G$ and conversely.  The set of such elements $x$ is called the \emph{center}
of $G$: The center is the subset of $G$ defined by:
$$Z(G) = \{x \in G \mid xa = ax \text{ for all a in } G\}.$$
\item[d.]  Show that the center of $G$ (as defined in part c) is a subgroup of $G$.  
\item[e.]  Let $x \in G$ be a fixed element and define $C_x = \{a \in G \mid x = axa^{-1}\}$.  Show that
$C_x$ is a subgroup of $G$ ($C_x$ is called the \emph{centralizer of $x$}).  
\item[f.] Let $G = \mathcal{S}(A)$ be the group of permutations of $A = \{a,b,c\}$.  Using the names for
the elements of this group we introduced in Problem Set 3, find all of the equivalence classes for the 
conjugacy relation on $G$ (there are three of them), determine the centralizer of each element of
$G$, and the center of $G$.  How are the sizes of the equivalence class of $x$ and the number of elements of the 
centralizer of $x$ related in each case? 
\end{enumerate}
\item[4.]  Let $H$ and $K$ be subgroups of a group $G$.
\begin{enumerate}
\item[a.]  Show that $H \cap K$ is also subgroup of $G$.
\item[b.]  Find an example where $H\cup K$ \emph{is} a subgroup of $G$ and one where $H\cup K$ \emph{is not}
a subgroup of $G$. 
\end{enumerate}
\end{enumerate}
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