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\centerline{Mathematics 351 -- Abstract Algebra 1}
\centerline{Discussion 2 -- Examples of Groups}
\centerline{October 19, 2007}
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\noindent
{\bf Background}
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We will now turn to a class of algebraic structures with 
just one operation, the {\it groups} you began the study of 
in the Algebraic
Structures class. 
We want to recall the definition
and study some properties and
examples of groups of various kinds.
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\noindent
{\it The definition}
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Recall that a group $(G,*)$ is a set with a binary operation $*$
satisfying:
\item{1)}  For all $a,b \in G$, $a * b\in G$.
\item{2)}  The operation $*$ is associative:
for all $a,b,c \in G$, $(a*b)*c = a*(b*c)$.
\item{3)}  There is an identity element $e \in G$ for $*$, satisfying
$a*e = a = e*a$ for all $a\in G$.  
\item{4)}  For each $a \in G$, there is an inverse of $a$ in $G$ for
the $*$ operation, satisfying $a*a^{-1} = e = a^{-1}*a$.  
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\noindent
{\it Discussion Questions}
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\item{A)} Let $R$ be a ring with addition and multiplication operations
$+$, $\cdot$.  Is $(R,+)$ a group?  Is $(R,\cdot)$ always a group?
\item{B)}  Show that if $R$ is a ring with identity and $U$ is the set of units in $R$,
then $(U,\cdot)$ is a group.
\item{C)} If the underlying set $G$ of a group is
{\it finite\/}, then it is possible to describe the 
operation $*$ completely by giving a {\it group operation 
table\/} with rows labeled by the elements $g \in G$, 
and columns also labeled by the elements $h \in G$, and
the entry in row $g$ and column $h$ equal to $g * h$.  
Show using the definition that each row and
each column of the group table of a finite group consists
of all the elements of $G$ in some order.  (That is, all the 
elements of $G$ appear, and each appears only once.)
\item{D)}  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 $G$ of invertible $2 \times 2$ matrices containing  $A$?  
Explain, and give the group operation table of $G$.
\item{E)}  Let $H$ be the set of six $3\times 3$ matrices 
given below:
$$\eqalign{I = \pmatrix{1&0&0\cr 0&1&0\cr 0&0&1\cr}, 
           S&= \pmatrix{0&0&1\cr 1&0&0\cr 0&1&0\cr},
           T = \pmatrix{0&1&0\cr 0&0&1\cr 1&0&0\cr},\cr
           U = \pmatrix{1&0&0\cr 0&0&1\cr 0&1&0\cr},
           V&= \pmatrix{0&0&1\cr 0&1&0\cr 1&0&0\cr},
           W = \pmatrix{0&1&0\cr 1&0&0\cr 0&0&1\cr}.\cr}$$ 
Is $H$ a group under the operation of {\it matrix 
multiplication\/}?  If so, give the operation table.
If not, say why not.
\item{E)}  Let $K$ be the set of six functions:
$$K = \{x,{1\over x},1-x,{1\over 1-x},{x-1\over x},
{x \over x-1}\}.$$
Is $K$ a group under the operation of {\it function 
composition\/}?  If so, give the operation table.
If not, say why not.
\bye
