**Assignment #1**(Due Wednesday, August 31**in class**)

Exercises**Assignment #2**(Due Wednesday, September 7**in class**)

Exercises**Assignment #3**(Due Wednesday, September 14**in class**)

Exercises**Assignment #4**(Due Wednesday, September 21**in class**)

Exercises**Assignment #5**(Due Wednesday, October 5**in class**)

Exercises [corrected 9/28]**Assignment #6**(Due Wednesday, October 12**in class**)

Exercises**Assignment #7**(Due Wednesday, October 19**in class**)

Exercises [clarified 10/16] please report any further issues**Assignment #8**(Due Wednesday, October 26**in class**)

Exercises please report any issues**Assignment #9**(Due Wednesday, November 9**in class**)

Exercises please report any issues**Assignment #10**(Due Wednesday, November 16**in class**)

Exercises Updated 11/10: problem 7 is wrong.**Assignment #11**(Due Wednesday, November 30**in class**)

Exercises Typo in #1 fixed on 11/24. Please report any further issues**Assignment #12**(Due Wednesday, December 7**in class**)

Exercises (typo in 3b fixed 12/05) Please report any issues

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