Math 200a Fall 2012

Instructors:

Professor:

Name Office E-mail Phone Office Hours Lecture Time Lecture Place
Prof. Daniel Rogalski AP&M 5131 drogalsk@math.ucsd.edu 534-4421 M 2-3pm, W 11am-12pm MWF 12-12:50pm AP&M 5402

Teaching Assistants:

Name Office E-mail Office Hours
Johanna Hennig 5132 AP&M jhennig@ucsd.edu TuTh 2:30-3:30pm

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

Below, we will post shot summaries of what was covered in each lecture. This list will not generally include advance information about upcoming lecture topics, so please ask me if you want to know what we will cover next.

• Lecture 1 (9/28/12): 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 (10/1/12): Chapters 1-3. Review of basic group theory II: cosets, Lagrange's theorem, normal subgroups, factor groups, composite groups, homomorphism theorems.
• Lecture 3 (10/3/12): Chapters 1-3, Section 6.3. Review of basic group theory III: orders of elements, cyclic groups. The free group F(X), presentations.
• Lecture 4 (10/5/12): Section 6.3, 4.1. Finish up proof of presentation of the dihedral group. 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 (10/8/12): 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. The conjugation action and conjugacy classes. The class equation. A p-group has a nontrivial center, p a prime.
• Lecture 6 (10/10/12): Section 4.3. A group of order p^2 is abelian (and either cyclic or a direct product of two cyclic groups of order p.) Review of S_n and A_n, including sketch of proof that the definition of even/odd permutation makes sense. Conjugacy classes in S_n correspond to cycle shapes. The number of conjugacy classes is the number of partitions of n.
• Lecture 7 (10/12/12):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/nZ) is isomorphic to (Z/nZ)^*. Using presentations to help compute automorphism groups, in particular showing that |Aut(D(n))| = n phi(n). The automorphism group of the direct product Z/pZ x Z/pZ is isomorphic to GL(2, Z/pZ), p a prime.
• Lecture 8 (10/15/12): Section 4.5-4.6 Proof that A_n is simple (method of proof from Hungerford.) Statement of the Sylow theorems.
• Lecture 9 (10/17/12): Section 4.5. Proof of the Sylow theorems.
• Lecture 10 (10/19/12):Section 4.5. Applications of the Sylow theorems to finding Sylow subgroups. Examples where you count elements. Infinite direct products. Recognizing direct products. If all Sylow subgroups of a group are normal, then the group is isomorphic to the direct product of its Sylow subgroups.
• Lecture 11 (10/22/12): Section 5.5. Semidirect products. Definition. Proof of group axioms. Recognition theorem for semidirect products.
• Lecture 12 (10/24/12): Section 5.5. Examples of Semidirect products. Classifying groups of low order using semidirect products.
• Lecture 13 (10/26/12): Section 6.1: Commutators and the commutator subgroup. Solvable groups. The derived series. Subgroups, factor groups of solvable groups are solvable, and extensions of two solvable groups are solvable.
• Review Session (10/29/12)
• Midterm (10/31/12)
• Lecture 14 (11/2/12): Guest lecture: Professor Salehi Golsefidy
• Lecture 15 (11/5/12): Section 6.1: Nilpotent groups. Central series. Normalizers grow in nilpotent groups. A finite group is nilpotent if and only if all of its Sylow subgroups are normal (and thus it is the direct product of these Sylow subgroups). The lower central series. The upper central series.
• Lecture 16 (11/7/12): Section 7.1-7.2. Rings. Basic definitions and examples. Matrix rings.
• Lecture 17 (11/9/12): Section 7.1-7.3. Polynomial rings. Group rings. Left and right ideals. Factor rings. Homomorphisms and the first isomorphism theorem.
• Veterans Day (No Class) (11/12/12):
• Lecture 18 (11/14/12): Section 7.4. Ideal generated by a subset. Maximal and prime ideals and characterizations in terms of factor rings. Examples of maximal and prime ideals.
• Lecture 19 (11/16/12): Section 7.4, 7.6. Posets and Zorn's lemma. Proof that every commutative ring with 1 has a maximal ideal, using Zorn's lemma. Proof that the nilradical of a commutative ring is the intersection of all prime ideals of the ring. Statement of the Chinese Reminder Theorem.
• Lecture 20 (11/19/12): Section 7.5, Section 15.4: Localization of a commutative ring at any multiplicative system. Statement of the universal property satisfied by the localization.
• Lecture 21 (11/21/12): Section 7.5, Section 12.1. Some examples of localization. The special cases of localizing at the powers of an element and localizing at a prime ideal. A localization at a prime ideal gives a local ring. The noetherian property for rings. Equivalent characterizations of the noetherian property.
• Thanksgiving Holiday (No Class) (11/23/12)
• Lecture 22 (11/26/12):Section 8.1. Euclidean domains. Examples: Z, F[x], Z[i]. A Euclidean domain is a PID.
• Lecture 23 (11/28/12):Section 8.2. UFDs. A PID is a UFD.
• Lecture 24 (11/30/12):Section 8.3. Rings of quadratic integers. Examples which are not UFDs. Factorization in Z[i] and sums of two squares.
• Lecture 25 (12/3/12):Proof that if R is UFD, so is R[x]. Gauss's lemma.
• Lecture 26 (12/5/12):Irreducible polynomials. Factor and remainder theorem. Fundamental theorem of algebra. Irreducible polynomials over R and C. Rational root theorem. Eisenstein criterion and examples.
• Lecture 27 (12/7/12):Proof of the Eisenstein criterion. A finite subgroup of the multiplicative group of a field is cyclic, in particular the units group of Z_p is cyclic. Also, the units group of Z_{p^n} is cyclic. Statement of the Hilbert basis theorem.

Homework

Exams