Basic course description

Math 200a is the first quarter of UCSD's three-quarter graduate-level abstract algebra course. At least one course at the undergraduate level in abstract algebra covering some group and ring theory is a prerequisite. The main aim of the course is to give PhD and masters students in mathematics sufficient background for their further studies, and to prepare these students for the qualifying exam in algebra given in May 2017 and again in September 2017. The text will be Dummit and Foote, "Abstract Algebra", 3rd edition.

Please follow the links at the right to read the syllabus and to find the homework assignments.

TA and Professor Contact Information

Professor Rogalski: office 5131 AP&M, e-mail drogalski@ucsd.edu.

• Lecture: MWF 11am-11:50am, 5402 AP&M
• Office hours: M 4-5pm and W 3-4pm in 5131 AP&M

TA: Francois Thilmany: e-mail fthilman@ucsd.edu

• Office hours: Tu 4-5pm and Th 3:30-4:30pm in 6132 AP&M (or in nearby seminar room 6218 AP&M)

Schedule of lectures

We plan to cover most of Chapters 1-9 of the text. The following schedule will be updated as we go with information about what we covered on particular days.
• 9/23/16 Chapters 1-3. Review of basic group theory I: definition of a group. Basic examples of groups: the symmetric group of a set, groups of numbers under +, multiplicative groups of numbers, the matrix group GL_n(F) for a field F, quaternion group, dihedral groups, integers mod n. Subgroups.
• 9/26/16 Chapters 1-3. Review of basic group theory II: homomorphisms and isomorphisms. Cosets and Lagrange's theorem. Normal subgroups and factor groups. The 1st isomorphism theorem.
• 9/28/16 Chapters 1-3. Review of basic group theory III: The 2nd-4th isomorphism theorems. Subgroup generated by a subset. Cyclic subgroups and cyclic groups. All cyclic groups are isomorphic to Z or Z/nZ.
• 9/30/16 Section 6.3. The free group on a set X. The universal property of a free group. Presentations. Example: the dihedral group D_{2n} is presented by two generators a, b with relations a^n = e, b^2 = e, ba = a^{-1} b.
• 10/3/16 More on presentations. The presentation of the dihedral group helps to compute endomorphisms of D_{2n}. It is hard to tell if a presented group is trivial or not. Section 4.1. Group actions. Basic examples of actions.
• 10/5/16 More on group actions. Orbits and the orbit stabilizer theorem. The action on left cosets. Application: a subgroup H of index equal to the smallest prime dividing |G| is normal. Burnside's counting lemma.
• 10/7/16 Example of burnside counting. Review of the symmetric group S_n. Every element of S_n is a product of disjoint cycles in an essentially unique way. Conjugacy classes in S_n correspond to distinct cycle shapes. The number of conjugacy classes is given by the partition function.
• 10/10/16 The alternating group A_n. Proof that A_n is simple for n at least 5.
• 10/12/16 Automomorphisms. Inner and outer automorphisms. Inn(G) is isomorphic to G/Z(G). Examples of calculating automorphism groups using presentations. Automorphisms of Z_n. Automorphisms of Z_p times Z_p as matrix groups.
• 10/14/16 Statement and proof of the Sylow theorems.
• 10/17/16 Applications of the Sylow theorems. Groups of order pq and pq^2. Using counting methods to show a group must have at least one normal Sylow subgroup. Characteristic subgroups. If H is characteristic in K and K is normal in G, then H is normal in G. Groups of order 105 have normal Sylow 5 and Sylow 7.
• 10/19/16 Recognizing internal direct products. A group with all of its Sylow subgroups normal is an internal direct product of its Sylow subgroups. Fundamental theorem of Abelian groups (no proofs).
• 10/21/16 Semidirect products. Recognizing semidirect products. Examples: D_{2n} is a semidirect product of Z_n and Z_2; all groups of order pq are semidirect products.
• 10/24/16 More examples of semidirect products. Using semidirect products to classify groups of low order.
• 10/26/16 Commutators and the commutator subgroup. Solvable groups. S_n is solvable only for n less than or equal to 4. The derived series. Subgroups and factor groups of solvable groups are solvable. If N is normal in G and G/N and N are solvable, then G is solvable.
• 10/28/16 in-class midterm
• 10/31/16 Nilpotent groups. Examples. Central series. The upper central series. Finite p-groups are nilpotent. Normalizers grow in Nilpotent groups. A finite nilpotent group is the same as a finite group with all of its Sylow subgroups normal, i.e. such that the group is the direct product of its Sylow subgroups. The lower central series.
• 11/2/16 Frattini's argument. A finite group is nilpotent if and only if every maximal subgroup is normal. Big theorems about solvable groups we won't prove. Introduction to rings: Definitions, basic examples, matrix rings, the zero ring.
• 11/4/16 More on rings: polynomial rings, group rings, direct products of rings. Degree in a polynomial ring. If R is a domain, so is R[x]. The group ring CG of a finite group G over the complex numbers C is isomorphic to a product of matrix rings over C (no proof). Ideals, factor rings. First isomorphism theorem for rings.
• 11/7/16 3rd and 4th isomorphism theorems. Examples of ideals. Zorn's lemma. Using Zorn's lemma to prove that every ring has a maximal ideal.
• 11/9/16 Localization. Multiplicative systems X in a ring R. The definition of the localization RX^{-1}. Universal property of the localization. Examples.
• 11/11/16 Veteran's Day Holiday (no class)
• 11/14/16 Prime and Maximal ideals. An ideal P is prime if and only if R/P is a domain, and an ideal M is maximal if and only if R/M is a field. Maximal ideals are prime. Examples of prime and maximal ideals. Two special cases of localization: localizing at the powers of an element f, and localizing at the complement of a prime ideal.
• 11/16/16 Chinese remainder theorem. Euclidean domains. Z and F[x], F a field, are Euclidean domains.
• 11/18/16 Quadratic integer rings. Z[i] is Euclidean. PIDs. A Euclidean domain is a PID, and moreover a nonzero ideal is generated by any element of minimal norm among nonzero elements in the ideal. Gcds.
• 11/21/16 Associates, irreducible, and prime elements. UFDs. prime elements are irreducible and the converse holds in a PID. Noetherian rings. PIDs are noetherian. Characterizations of noetherian rings.
• 11/23/16 Proof that a PID is a UFD, in fact, every noetherian domain such that irreducible elements are prime is a UFD. In a UFD, gcd's exist. In a UFD, irreducible elements are prime. Examples. Multivariate polynomial rings. R[x] is not a PID if R is a PID which is not a field, but R[x] is a UFD for any UFD R (next time)
• 11/25/16 Thanksgiving Holiday (no class)
• 11/28/16
• 11/30/16
• 12/2/16
• Tu 12/6/16 Final Exam 11:30am-2:30pm