Math 200a Fall 2011

Announcements:

The final Exam is Wednesday 12/7/2011 from 3-6pm in our usual class room. Office hours between now and then are by appointment. The exam covers everything we have done except the final lecture's topic (the Hilbert basis theorem).

Instructors:

Professor:

Teaching Assistants:

Course description:

This is the first quarter of the three-part graduate algebra sequence. In this quarter, we will cover primarily group and ring theory as in Chapters 1-9 of Dummit and Foote's text, Abstract Algebra. Please find the more detailed syllabus for the class, including information about homework, exams, academic honesty, etc., below:

Math 200a course syllabus

Lecture Summaries

• Lecture 1 (9/23/11): Chapters 1-3. Review of basic group theory I: groups, subgroups, examples such as S_n, D_{2n}, Q_8, GL_n(F), (Z/nZ, +), (Z/nZ)*, homomorphisms.
• Lecture 2 (9/26/11): Chapters 1-3. Review of basic group theory II: cosets, Lagrange's theorem, normal subgroups, factor groups, composite groups, homomorphism theorems.
• Lecture 3 (9/28/11): Chapters 1-3, Section 6.3. Review of basic group theory III: orders of elements, cyclic groups. The free group F(X), presentations, examples.
• Lecture 4 (9/30/11): Section 6.3, 4.1. Finish up presentations. Begin group actions: definition, examples, equivalence of actions of G on X and homomorphisms G to permutation group of X, Cayley's theorem.
• Lecture 5 (10/3/11): Section 4.1-4.2 Orbits of group actions and orbit/stabilizer theorem. Action of G on left cosets of a subgroup H. Application: if p is the smallest prime dividing |G| then any subgroup of index p is normal in G. Burnside's counting formula (also called Cauchy/Frobenius formula) for the number of orbits of an action (not in text).
• Lecture 6 (10/5/11): Section 4.3. Application of counting formula to count number of necklaces of black and white beads, up to rotation equivalence (not in text). The conjugation action and conjugacy classes. The class equation. A p-group has a nontrivial center, p a prime. A group of order p^2 is abelian (and either cyclic or a direct product of two cyclic groups of order p.) Conjugacy classes in S_n. The number of conjugacy classes is the number of partitions of n.
• Lecture 7 (10/7/11):Section 4.4. Automorphisms and the automorphism group Aut(G). The inner automorphism group Inn(G) is isomorphic to G/Z(G). Examples of inner automorphism groups. Aut(Z_n) is isomorphic to (Z_n)^*. Using presentations to help compute automorphism groups, in particular showing that |Aut(D(n))| = n phi(n).
• Lecture 8 (10/10/11):Section 4.6. The alternating group A_n. Proof that A_n is simple for n at least 5. Statement of the Sylow Theorems.
• Lecture 9 (10/12/11):Section 4.5. Proof of the Sylow Theorems.
• Lecture 10 (10/14/11):Section 4.5, 5.1, 5.3. Applications of the Sylow Theorems to finding normal subgroups of finite groups. Direct products of collections of groups. Theorem about recognizing a group as an internal direct product of finitely many subgroups.
• Lecture 11 (10/17/11):Section 5.5. Semidirect products. Recognition theorem for semidirect products.
• Lecture 12 (10/19/11):Section 5.5. Examples of semidirect products. Classifying groups in terms of semidirect products. Classification of groups of order pq. Classification of groups of order 12.
• Lecture 13 (10/21/11):Section 5.4, 6.1. Commutators and the commutator subgroup. Nilpotent groups and various characterizations of them. The upper central series.
• Lecture 14 (10/24/11): Section 6.1. More on nilpotent groups. Normalizers grow in nilpotent groups. A finite group is nilpotent if and only if all of its Sylow subgroups are normal. Subgroups and factor groups of nilpotent groups are nilpotent. Lower central series. Sovable groups. The derived series.
• Lecture 15 (10/26/11): Section 6.1, Section 3.4. Finish solvable groups. Subgroups and factor groups of solvable groups are solvable. If N and G/N are solvable then so is G. Composition series. A finite group is solvable if and only if it has only cyclic groups of prime order as its composition factors. The Jordan-Holder theorem.
• Lecture 16 (10/28/11): Section 7.1-7.2. Introduction to rings. Definitions, basic properties and examples. Domains, fields, units. Matrix rings. Polynomials rings.
• Lecture 17 (10/31/11): Section 7.2-7.3. More basics on rings. Groups rings. Direct products. Subrings, ideals, factor rings, homomorphisms, isomorphism theorems.
• Midterm Exam (11/2/11)
• Lecture 18 (11/4/11): Section 7.4. Prime and maximal ideals, characterizations and examples. Zorn's lemma. The proof using Zorn's lemma that every ring contains a maximal ideal.
• Lecture 19 (11/7/11):Section 7.5 and 15.4. Localization. Basic definition, properties, examples.
• Lecture 20 (11/9/11):Section 7.5-7.6. A few more examples of localization. The Chinese remainder theorem.
• No class, Veteran's Day (11/11/11):
• Lecture 21 (11/14/11):Section 8.1, 9.2. Euclidean domains and examples: Z, F[x], Z[i]. A Euclidean domain is a PID and in particular any nonzero ideal is generated by an element of minimal norm in it.
• Lecture 22 (11/16/11):Section 8.1-8.2. GCD's exist in PID's. Prime, irreducible elements in domains. prime elements are irreducible and the converse holds in a PID. In a PID prime ideals are maximal and are exactly those ideals generated by irreducible elements. Noetherian rings. Equivalent characterizations of noetherian rings.
• Lecture 23 (11/18/11): Section 8.2-8.3, 9.1. A PID is a UFD; in fact proved a noetherian ring in which every irreducible is prime is a UFD. In a UFD, GCDs exist, and irreducible elements are prime. Polynomial rings in several variables. Examples of UFDs that are PIDS, and examples of non-UFDs.
• Lecture 24 (11/21/11):Section 9.3 Gauss's lemma: if R is a UFD with field of fractions F and a polynomial in R[x] factors in F[x], then after adjusting by scalars it factors in R[x]. If R is a UFD, then R[x] is a UFD.
• No class, Thanksgiving holiday (11/23/11), (11/25/11):
• Lecture 25 (11/21/11): Section 8.3. Factorization in quadratic integer rings. Z[sqrt(-5)] is not a UFD. Explicit calculation of the irreducible elements of the Gaussian integers Z[i].
• Lecture 26 (11/21/11):Section 9.4-9.5. Tests of irreducibility of polynomials, in particular the Eisenstein criterion. Any finite subgroup of the multiplicative group of a field is cyclic, so the multiplicative group of Z/pZ for a prime p is cyclic.
• Lecture 27 (11/21/11):Section 9.6. The Hilbert basis theorem.

Homework

• Homework 1: Homework 1 due Friday 9/30

• Homework 2: Homework 2 due Friday 10/7

• Homework 3: Homework 3 due Friday 10/14

• Homework 4: Homework 4 due Friday 10/21

• Homework 5: Homework 5 due Friday 10/28

• Homework 6: Homework 6 due Friday 11/11

• Homework 7: Homework 7 due Tuesday 11/22

• Homework 8: Homework 8 due Friday 12/2

• Old exam from 2008 (ignore the problem about categories): Old midterm
  Also, you can see the 2010 midterm posted on Professor Oprea's website for Math 200a in Fall 2010. Ignore the problem about rings.

• Old final exam from fall 2008 (Only the problems 1-3 on groups are relevant): Old final
  Also, you can see the final posted on Professor Oprea's website for Math 200a in Fall 2010. This exam is longer than ours will be.