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\centerline{Mathematics 243 -- Algebraic Structures}
\centerline{Discussion 3 -- Examples of Groups}
\centerline{November 8, 2006;  Writeups due: Monday, November 13}
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{\it Background}
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In the last two classes, we have 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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I.  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!) 
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\noindent
II.
Let $K$ be the set of six functions (with domain
${\bf R} - \{0,1\}$):
$$K = \left\{f_1(x) = x,f_2(x) = {1\over 1-x},f_3(x) = {x-1\over x},
f_4(x) = {1\over x}, f_5(x) = 1-x,
f_6(x) = {x \over x-1}\right\}.$$
$K$ 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. 
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\noindent
III.  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 matrix multiplication -- multiplying
any two matrices in $H$ must produce a matrix in the same set.
So in particular, $A \cdot A = A^2, A^3, \ldots, A^{-1}, \ldots$ 
must also be in $H$.  Give the group operation table of $H$.
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{\it over for question IV}
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IV. 
\item{A)} Comparing the example groups $G,H$, and $K$ from questions
I, II, and III, 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 Monday that just saying the two groups have the same
order (same number of elements) is not sufficient. 
\item{B)} Each of the groups in questions I,II,III contains 
several (proper) subsets that are groups in their own right under the given 
operation.  Find at least three such subsets in each case.
\bye
