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\centerline{Mathematics 243, section 1 -- Algebraic Structures}
\centerline{Exam 3 -- December 1, 2006}
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\noindent
{\it Directions}
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Do all work in the blue exam booklet.  There are 100 possible 
points.
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\noindent
I.  In an RSA public key cryptosystem, the public key information
is $m = 323$ and $e = 11$.  Messages consisting of capital roman
letters and blanks are encoded as 3-digit blocks $000, 001, \cdots, 026$
(with $A = 000$, $B = 001$, ... , $Z = 025$, and blank = $026$)
and encrypted as 3-digit blocks.
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\item{A)}  (15) How would the letters $US$ be encrypted?
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\item{B)}  (10) What is the (secret) decryption function $g$?
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II. (20) Let
$$H = \left\{A = \pmatrix{a & 0 \cr b & c\cr} : a,b,c \in {\bf R}\ {\rm and}\ a, c \ne 0\right\}$$
Is $H$ a group under the operation of matrix multiplication?
If so, give a proof.  If not, say which of the group properties fail.
{\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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\noindent
III.  Let $G$ be a group, let $a \in G$ be a fixed element, and 
consider the mapping $\ell_a : G \to G$ defined by $\ell_a(x) = ax$ for 
all $x \in G$.  
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\item{A)}  (10) Show that $\ell_a$ is one-to-one and onto. 
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\item{B)}  (5)  What is the inverse mapping of $\ell_a$? 
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IV. 
\item{A)} (15) Let $G$ be a cyclic group with generator $a$.  Show
that every subgroup of $G$ is also cyclic.
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\item{B)}  (10) $G = {\bf Z}_{30}$ is a group under the operation of addition mod 30.
Find {\it all of} the generators of the cyclic subgroup 
$H = \langle [18] \rangle$.  
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V.  Let $G,H$ be two groups.  
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\item{A)} (5)  What is the definition of a group homomorphism
from $G$ to $H$?
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\item{B)} (10) Let $G = {\bf Z}_{24}$ with operation addition
mod 24.  What is the kernel of the group homomorphism
$\phi : G \to G$ defined by $\phi([x]) = [9x]$?
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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
%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 $G$?  
\vfill\eject
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{\it The General Pattern for $2\times 2$ Matrix Inverses}
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\noindent
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
$$A^{-1} = \pmatrix{ {d\over ad - bc} & {-b\over ad - bc}\cr
                     {-c\over ad - bc} & {a\over ad - bc}\cr}$$
\bye




