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\centerline{Mathematics 243 -- Algebraic Structures}
\centerline{Discussion 3 -- Examples of Groups}
\centerline{October 31, 2003; Writeups due: Friday, November 7}
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{\it Background}
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In Wednesday's class, we introduced the definition of a group and some 
first examples.  Today we want to continue and study some additional 
properties and examples of groups of various kinds.
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{\it Discussion Questions}
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\item{A)}  Show that if $(gh)^{-1} = g^{-1}h^{-1}$ for all $g,h$
in a group $G$, then $G$ is abelian (that is the group operation
is commutative). 
\item{B)}  Let $G$ be the set of six $3\times 3$ matrices 
given below:
$$\eqalign{I = \pmatrix{1&0&0\cr 0&1&0\cr 0&0&1\cr}, 
           S&= \pmatrix{0&0&1\cr 1&0&0\cr 0&1&0\cr},
           T = \pmatrix{0&1&0\cr 0&0&1\cr 1&0&0\cr},\cr
           U = \pmatrix{1&0&0\cr 0&0&1\cr 0&1&0\cr},
           V&= \pmatrix{0&0&1\cr 0&1&0\cr 1&0&0\cr},
           W = \pmatrix{0&1&0\cr 1&0&0\cr 0&0&1\cr}.\cr}$$ 
$G$ is a group under the operation of {\it matrix 
multiplication\/}.  Give the operation table, labeling
the rows and columns in this order.
(Write the entries of the table using the names $I,S,T,U,V,W$,
not the matrices!) 
\item{C)}  The $2 \times 2$ matrix  $A$ given by 
$$A = \pmatrix{ {1\over 2} & {-\sqrt{3} \over 2}\cr
              {\sqrt{3} \over 2} & {1\over 2}\cr}$$
is an invertible matrix (why?).  What is the {\it smallest\/} 
group $H$ of invertible $2 \times 2$ matrices containing  $A$?  
Hint: A group must be ``closed'' under its operation -- combining
any two elements using the operation must produce an element of the same set.
So in particular, $A \cdot A = A^2$ must also be in $H$.
Give the group operation table of $H$.
\item{D)}  Let $K$ be the set of six functions (with domain
${\bf R} - \{0,1\}$):
$$K = \left\{x,{1\over 1-x},{x-1\over x},{1\over x},1-x,
{x \over x-1}\right\}.$$
$K$ is a group under the operation of {\it function 
composition\/}.  Give the operation table, labeling the 
rows and columns by the functions in this order.  It may help
to abbreviate the functions by giving them names(!)
\item{E)}  Comparing the example groups $G,H$, and $K$ from questions
B, C, and D, if you 
look closely, you should start to realize that 
two of these are ``the same''.  In what way?  What should mean for
two groups to be ``the same''?  Recall that we saw in 
class on Wednesday, that just saying the two groups have the same
order is not sufficient. 
\bye
