427 homework assignments and lecture notes

Fall 2016, section G

(Last year version of the course, with lecture notes, is here. This year's version is different)

lecture 1 (08/22/16) Well-order and induction. Division algorithm in Z.
lecture 2 (8/24/16) greatest common divisors
lecture 3 (8/26/16) factorization into primes, infinity of primes
lecture 4 (8/29/16) equivalence relations, partitions, Z/nZ
lecture 5 (08/31/16) definition of a group
lecture 6 (9/02/16) the notions of a homomorphism and an isomorphism
lecture 7 (9/07/16) definition of a ring, a field, subgroups
lecture 8 (9/09/16) Cyclic subgroups. Kernel of a homomorphism.
lecture 9 (9/12/16) Kernel of a homomorphism measures injectivity, normal subgroups, permutation groups, cycles.
lecture 10 (9/14/16) Every permutation is a "unique" product of disjoint cycles.
lecture 11 (9/16/16) group actions, orbits, orbits define a partition
lecture 12 (9/19/16) Right and left cosets, Lagrange's theorem.
lecture 13 (9/21/16) quotient groups, homomorphism theorem
lecture 14 (9/23/16) classification of cyclic groups, Euler phi-function, products of groups
review problems for the first midterm (09/26/16)
first midterm (09/28/16)
lecture 15 (9/30/16) review of vector spaces, linear maps and matrices.
lecture 16 (10/03/16) Canonical representation of S_n on R^n. Cayley's theorem.
lecture 17 (10/05/16) Diheadral groups D_n. Semi-direct products.
lecture 18 (10/07/16) semi-direct products
lecture 19 (10/10/16) Semi-direct products 2.
lecture 20 (10/12/16) Definition of a ring, units, ring homomorphism, commutative ring, polynomial rings, rings of functions.
lecture 21 (10/14/16)"substitution principle," ideals, polynomials are not functions.
lecture 22 (10/17/16) quotient rings, homomorphism theorem.
lecture 23 (10/19/16) ideals generated by a set, zero divisors, integral domains, finite integral domains are fields
lecture 24 (10/21/16) division algorithm for polynomials over a field and some consequences.
lecture 25 (10/24/16) maximal and prime ideals.
lecture 26 (10/26/16) Euclidian rings are PIDs.
lecture 27 (10/28/16) Irreducibles and primes. In an an integral domain primes are irreducibles but there may be irreducibles that are not primes. In a PID irreducibles are the same as primes.
review problems for the second midterm (10/31/16)
second midterm (11/02/16)
lecture 28 (11/04/16) Irreducible polynomials with complex and real coefficients.
lecture 29 (11/07/16) Gauss Lemma. Factorizations in Q[x] and Z[x]. Modular irreducibility test.
lecture 30 (11/09/16)Unique factorization domains (UFDs). Ascending chains of ideals. In a PID ascending chains stabilize. Existence of factorization into irreducibles in a PID
lecture 31 (11/11/16) Uniqueness of factorization in PIDs. Sign representation of the symmetric group.

A few books on representation theory:
Linear Representations of Finite groups by J-P Serre (available on campus)
Quantum Theory, Groups and Representations: An Introduction by Peter Woit
Groups and Symmetries From Finite Groups to Lie Groups by Yvette Kosmann-Schwarzbach (available on campus)
Linear Representations of Groups by Ernest B. Vinberg (available on campus)

lecture 32 (11/14/16) Repersentations from actions on (finite) sets, direct sums
lecture 33 (11/16/16) Hermitian inner products, ...
lecture 34 (11/18/16) unitary representations are direct sums of irreducibles, representations of finite groups are unitary, tensor products of vector spaces.
lecture 35 (11/28/16) Tensor products of representations. Symmetric and alternating tensors.
lecture 36 (11/30/16) Symmetric and alternating tensors, trace, conjugacy classes, characters.
lecture 37 (12/02/16) properties of characters and some of their consequences
lecture 38 (12/05/16) Schur's lemma and orthogonality of characters.
Review problems (12/07/16) Please report any issues.
Last modified: Tue Dec 6 10:13:57 CST 2016