Math 120 - Modern algebra - Fall 2011

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 (D₆). 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 A₄. 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 A₅. 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].

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 (D₆). 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 A₄. 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 A₅. 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].