Mathematics 504: Abstract Algebra I Syllabus
- Fundamentals: sets, relations, and functions, Cartesian
products and operations, partial orderings, equivalence relations and
partitions, the Axiom of Choice and Zorn's Lemma, cardinal numbers.
- Integers: mathematical induction and the Well Ordering
Principle, congruence, Division Algorithm, unique factorization,
greatest common divisor and least common multiple, Euclidean Algorithm.
- Groups (basic theory): semigroups and monoids, various
characterizations of groups, subgroups, normal subgroups, homomorphism,
isomorphism, quotient groups, direct products and sums, cosets and
counting, Lagrange's Theorem, subgroup generation, Isomorphism
- Examples of groups: permutation groups, groups of symmetries, matrix groups, dihedral and quaternion groups.
- Permutation groups: Cayley's Theorem, permutations as
products of disjoint cycles and consequences for the structure of
permutation groups, permutations as products of transpositions,
alternating groups and their simplicity.
- Abelian groups: structure of cyclic groups, free abelian
groups and the structure of finitely generated abelian groups, the
Fundamental Theorem of Abelian Groups.
- Groups: structure theory, group actions on sets,
stabilizers, the class equation, generalizations of Cayley's Theorem,
Cauchy's Theorem, Sylow Theorems, classification of finite groups of
- Rings (basic theory): subrings, ideals, homomorphism,
isomorphism, quotient rings, direct products and sums, isomorphism and
correspondence theorems, division rings and fields, examples of rings,
rings of endomorphisms of an abelian group, rings of matrices.
- Rings (advanced theory): properties of ideals, maximal and
prime ideals, the Chinese Remainder Theorem, integral domains,
relationship between maximal ideals and fields and between prime ideals
and integral domains, factorization in commutative rings, irreducible
and prime elements, Euclidean domains, principal ideal domains, unique
factorization domains, polynomial rings.
Gain familiarity with basic structures and techniques of abstract algebra, in order to be able to develop them and apply them in other areas of mathematics.