Instructor: Dr. Áron BERECZKY
Text: notes taken in class
reference book: Peter J. Cameron: Introdction to Algebra, Oxford Univerity Press, 1998. (chapters 1, 2, 3, sections 7.1.1, 7.1.2, 7.2.1.)
The course provides an introduction to group theory and ring theory.
The course provides an introduction to ring theory and group theory.
Topics to be covered:
- Some prerequisites: functions, relations, congruence on Z.
- Group Theory: axioms, basic properties, isomorphism, order, subgroups, subgroup tests, cyclic groups, subgroups of cyclic groups, unit groups of rings, centralizers of elements, center, cosets of subgroups, Lagrangeâ€™s theorem, dihedral groups, permutations, cycle form, order and parity of permutations, alternating groups, Cayleyâ€™s theorem, Cauchyâ€™s theorem, centralizers of permutations and conjugacy in the symmetric groups, conjugacy relation in general, conjugacy classes, conjugacy of subgroups, normal subgroups (equivalent definitions), normalizers and centralizers of subgroups, factor groups, homomorphisms (basic properties, image and kernel), Isomorphism theorems, automorphism groups, direct products, inner direct product, structure of finite commutative groups (without proof), groups acting on sets, orbits, stabilizer subroups, groups of small oder, groups of order , Sylow theorems (without proof).
- Ring Theory: axioms, basic properties, basic notations, subrings, subring tests, ideals, congruence modulo ideals, congruence classes (=cosets), factor rings, simple rings, rings without nontrivial left ideals, homomorphisms (basic properies, image and kernel), Isomorphism theorems, maximal ideals, principal ideals, division in integral domains, units, associates, greatest common divisors, primes, irreducibles, Unique Factorization Domains, Principal Ideal Domains, Euclidean Domains and their connections (with examples), the ring of Gaussian Integers, fields of fractions, simple algebraic and transcendental extensions.