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\centerline{Mathematics 243, section 1 -- Algebraic Structures}
\centerline{Review Sheet, Final Exam}
\centerline{December 8, 2003}
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\noindent
{\it General Information}
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The final examination for this course will be given at 
8:30 a.m. on Tuesday, December 16 in our regular class room, 
Swords 302.
The exam will be roughly twice the length of one of the three
midterms, but you will have the full three hour period from
8:30 am to 11:30 am to work on it if you need that much time.
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\noindent
{\it Topics to be Covered}
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\item{1)}  Basic logic:  Logical connectives and truth tables; 
implications and their converses, contrapositives, and inverses; quantifiers
\item{2)}  Sets: set operations (union, intersection, difference, complement,
Cartesian product, etc.) and their properties.
\item{3)}  Mappings: the 1-1 and onto properties, direct and inverse
images of sets under mappings,
\item{4)} Binary operations: identity elements, inverses, properties
such as associativity, commutativity, key examples such as 
function composition, matrix addition and multiplication, 
addition and multiplication in ${\bf Z}$ and ${\bf Z}_n$, etc.  
\item{5)}  Relations:  especially equivalence relations and 
the partition of a set into equivalence classes under an 
equivalence relation (key example:  the congruence mod $n$
relation on ${\bf Z}$ -- the set of equivalence classes
in that case is ${\bf Z}_n$).
\item{6)}  Properties of ${\bf Z}$:  the Well-Ordering
Property, proof by mathematical induction, the division algorithm,
divisibility, prime numbers and prime factorizations,
the gcd and lcm of two integers, Euclid's algorithm for the
gcd.
\item{7)}  The congruence mod $n$ relation on ${\bf Z}$
and the integers modulo $n$,  addition and multiplication
in ${\bf Z}_n$ and their properties.
 Applications to cryptography (affine and RSA cryptosystems). 
\item{7)}  Groups:  the definition, key examples such as
${\bf Z}_n$ under addition, the group of units in ${\bf Z}_n$
under multiplication, $GL_2({\bf R})$, the symmetric groups
$S_n$, groups of symmetries of geometric figures, etc.
\item{8)}  Subgroups of groups:  know how to determine whether
a given subset of a group is a subgroup.  Cyclic subgroups.
\item{9)}  Cyclic groups, generators, orders of elements, etc.
\item{10)}  Homomorphisms of groups, kernel and image of a 
homomorphism, isomorphisms.
\item{11)}  Cayley's Theorem -- know how to find a 
permutation group isomorphic to a given finite group.
\item{12)}  Cosets of a subgroup and Lagrange's Theorem
for finite groups -- know the statement: {\it If 
$G$ is a finite group and $H$ is a subgroup, then
$o(H) | o(G)$}.  Also know what it says about orders 
of elements in a finite group.
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{\it Proofs to Know}
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See the review sheets for Exams 2 and 3
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\noindent
{\it Suggestions on How to Study}
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Start by reading the above list of topics carefully.
If there are terms there that are unfamiliar or for which
you cannot give the precise definition, {\it learn the definitions
now; memorize them if you have to!}  You simply cannot answer
a question about this material if you do not know what the 
terms mean.  Review the class notes.  {\it Everything on 
the final will be similar to something we have discussed at
some point this semester.}  Also look back over your graded
problem sets and exams.  If there are problems that you 
did not get the first time around, try them again now.  Then
go through the suggested problems from the three
review sheets.  If you have worked these out previously,
it is not necessary to do them all again.  But try a representative
sample ``from scratch'' -- don't just look over your old
solutions and nod your head if it looks familiar.  You
need the practice thinking through the logic of how
the solution is derived again!
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\noindent
{\it Review Session}
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I will be happy to run a review session for the final
exam during study week.  We can discuss a time in class
on Monday, December 8.
\bye

