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\centerline{Mathematics 243, section 1 -- Algebraic Structures}
\centerline{Exam 3 -- December 5, 2003}
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{\it Directions}
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Do all work in the blue exam booklet.  There are 100 possible regular
points and 10 Extra Credit points.
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I.  Let $G$ be a group, let $a \in G$ be a fixed element, and 
consider the mapping $h_a : G \to G$ defined by $h_a(x) = xa$ for 
all $x \in G$.  
\item{A)}  (10) Show that $h_a$ is one-to-one and onto.  
\item{B)}  (10) If $G$ is a finite group (that is, $o(G)$ is finite),
what does the result from part A say about the group operation 
table of $G$?
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II. 
\item{A)}  (20) In the group $G = {\bf Z}_{24}$,
in which the operation is addition mod 24,
find {\it all of} the generators of the cyclic subgroup 
$H = \langle [15] \rangle$.  
\item{B)}  (20)  Write the
element $\sigma = (1532)(24368)(12479)$ in the permutation group $S_9$
as a product of disjoint cycles and determine the order of $\sigma$.
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III.  Let $\phi : G \to H$ be a group homomorphism.
\item{A)} (10) Show that if $e_G$ and $e_H$ are the identity elements in 
the groups $G$ and $H$, then $\phi(e_G) = e_H$.
\item{B)} (10) Show that $\ker(\phi)$ is a subgroup of $G$.
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IV.  Let $G$ be the set of all $2\times 2$ matrices with real entries
of the form $A = \pmatrix{a & b\cr -b & a\cr}$, where $a,b$ are
any real numbers, {\it at least one of which is not zero}.
\item{A)}  (10) Is $G$ closed under the operation of matrix multiplication?
\item{B)}  (10) Does every element of $G$ have a (multiplicative) inverse
which is an element of $G$?  {\it Note:  We discussed a general rule
for finding the inverse of a $2\times 2$ matrix in class.  If you don't
remember it, you can ``buy'' it from me during the exam in return 
for a reduction of 5 points on your score for this problem.}
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{\it Extra Credit} (10) Give an example of an invertible $2\times 2$ 
matrix that has order 7 as an element of the group 
$GL_2({\bf R})$.
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{\it The General Pattern for $2\times 2$ Matrix Inverses}
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If 
$$A = \pmatrix{a & b \cr c & d\cr}$$
then $A$ has a inverse matrix (for multiplication) if and only 
if $\det(A) = ad - bc \ne 0$, and then
$$A^{-1} = \pmatrix{ {d\over ad - bc} & {-b\over ad - bc}\cr
                     {-c\over ad - bc} & {a\over ad - bc}\cr}$$
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