Lecture plan
Date | Covered |
---|---|
9/27 | Welcome. Motivating example: the group of symmetries of a cube in R^3. Axioms of a group (a set and a binary operation, which is associative, has a unit, and where all elements have inverses). Immediate consequences of the axioms. Order of an element in a group. Order of a (finite) group. The dihedral group of order 2n. Beginning of symmetric groups. |
9/29 | Symmetric groups: cycles, cycle decomposition. Matrix groups. Homomorphisms and isomorphism. |
10/4 | Group actions (much more on that in chapter 4). Group action gives a homomorphism into a symmetric group. Subgroups. Concrete examples of subgroups. Abstract examples of subgroups: Kernel, Image, Centralizer, Center. |
10/6 | More on subgroups and group actions: Normalizer, stabilizer. Cyclic groups and cyclic subgroups. |
10/11 | Subgroups generated by a subset. Lattice of subgroups. Quotient groups: Necessary condition for a subgroup to be the kernel of a homomorphism, left cosets gN of a subgroup N of G. G/N as the set of left cosets. |
10/14 | Normal subgroups. Group structure on G/N when N is normal. Criteria for normality. Natural homomorphism from G to G/N. First isomorphism theorem. A homomorphism is injective if and only if its kernel is trivial. Example: R/Z is isomorphic to the unit complex numbers. |
10/18 | Another example: GL(n,C)/SL(n,C) is isomorphic to C-{0}. Lagrange's theorem and many corollaries. Index of a subgroup. A subgroup of index 2 is normal. The symmetric group is generated by transpositions. The sign homomorphism from the symmetric group to {+1,-1} and how to calculate it (m-cycles have sign (-1)^(m-1)). The alternating group. Simple groups. |
10/20 | The story of classification of finite simple groups. More on group actions: orbits and their relation to stabilizers. Permutation groups (subgroups of symmetric groups). Cayley's theorem: any group is isomorphic to a permutation group. A subgroup of a finite group G whose index is the smallest prime dividing |G| is normal. The conjugation action and the class equation. Groups whose order is a power of a prime have non-trivial center. |
10/25 | Midterm exam. |
10/27 | Example of the class equation (D_{6}). Groups of order the square of a prime are abelian. Cauchy's theorem (following the proof outlined in exercise 9, page 96). Conjugation of m-cycle in S_{n}. Two permutations are conjugate if and only if they have the same cycle type. A_{n} is generated by 3-cycles. |
11/1 | A_{n} is simple for n at least 5. Decomposition series and solvable groups (section 3.4). S_{n} is solvable if and only if n < 5. Statement of Sylow's theorem. |
11/3 | Proof of Sylow's theorems. Examples: Sylow subgroups of A_{4}. Groups of order 12. Groups of order pq for two different primes p, q. |
11/8 | Groups of order 60: a simple such group is isomorphic to A_{5}. Rest of the class was various loose ends about groups. Homomorphisms out of a simple group are either trivial or injective. Direct products and semidirect products: definition and recognition theorems. Classification of groups of order pq. |
11/10 | Rings: Axioms, basic properties and examples. Units and zero divisors. Fields and integral domains. Cancellation in integral domains. Subrings. Polynomials rings R[x]. Degrees of polynomials, units in polynomial rings, R[x] is an integral domain if R is. |
11/15 | Ring homomorphisms and isomorphisms. Ideals. Quotient rings and the first isomorphism theorem. Generators for ideals. Example: R[x]/(x^2+1) is isomorphic to C. |
11/17 | Principal ideals. Fourth isomorphism theorem. Maximal ideals. Prime ideals. Sum, product, intersection of ideals. |
11/29 | Chinese remained theorem. Euclidean domain. Z and polynomial rings over a field are Eculidean domains. PID. UFD. |
12/1 | Irreducible and prime elements in an integral domain. Prime implies irreducible. In a PID, irreducible implies prime. Example R=Z[sqrt(-5)]: 3 is irreducible but not prime. PID implies UFD. Z[sqrt(-5)] is not a UFD, in fact 9 has non-unique factorization into irreducibles. The Gaussian integers Z[i] is a Euclidean domain. |
12/6 | Each prime number p in Z congruent to 3 mod 4 is also irreducible in Z[i]. Each prime congruent to 1 mod 4 can be written as the sum of two squares and splits as the product of two irreducibles in Z[i]. |