# Math 200a Fall 2014

## The Final exam is Monday 12/15, 11:30am-2:30pm in our usual room. A sample final (from fall 2011) has been posted below the homework at the bottom of the page. Ignore problem 6; localization and the noetherian property were both studied more extensively in the fall that year. (We will study them more in Math 200c).

### 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 11-11:50am, W 2-2:50pm MWF 1-1:50pm AP&M 5402

#### Teaching Assistant:

Name Office E-mail Office Hours
Rob Won AP&M 6321 rwon@ucsd.edu Tu 1-1:50pm, Th 1-1:50pm

Course Description and Syllabus:

Math 200a is the first quarter of the three-part graduate algebra sequence. In this quarter, we will cover primarily group theory and basic ring theory as in Chapters 1-9 of Dummit and Foote's text, Abstract Algebra. Math 200b will concentrate on the theory of modules and fields, while Math 200c will cover the more advanced theory of commutative rings. The detailed syllabus for the course can be found at the following link. It is important that all students read it since the full course policies may not be gone over in class. Please ask if you have any questions about these policies.

### Math 200a Fall 2014 course syllabus

Lecture Summaries

Below, we will post short summaries of what was covered in each lecture.

• (10/3/14): NO CLASS
• Lecture 1 (10/6/14): (Chapters 1-3) Review of basic group theory: definition of group, subgroups, groups of numbers, symmetric groups, dihedral group, the quaternion group, integers mod n. Homomorphisms and isomorphisms.
• Lecture 2 (10/8/14): (Chapters 1-3) left and right cosets, Lagrange's theorem, normal subgroups, factor groups, 1st isomorphism theorem, group generated by a subset, structure of cyclic groups.
• Lecture 3 (10/10/14): (Sections 3.2, 3.3) Composites. Formula for |HK|. HK is a subgroup if and only if HK = KH. Definition of normalizer. HK is a subgroup if H is contained in the normalizer of K. 2nd, 3rd, and 4th isomorphism theorems. Examples using (Z, +).
• Lecture 4 (10/13/14): (Sections 1.3, 1.7, 3.5, 4.1) The symmetric group. Disjoint cycle form of a permutation. Transpositions and the alternating subgroup A_n. Definition of group actions and simple examples.
• Lecture 5 (10/15/14): (Sections 4.1, 4.2) Actions of G on X are the same as homomorphisms from G to the symmetric group on X. Cayley's theorem. Orbits of an action. The orbit-stabilizer theorem and examples. G acts on left cosets of a subgroup H. A subgroup H of index p, where p is the smallest prime dividing |G|, is normal.
• Lecture 6 (10/17/14): (Section 4.3, 4.4) More on the action of G on itself by conjugation. The class equation. p-groups have non-trivial center. Groups of order p^2 are Abelian. Each conjugacy class in S_n consists of all permutations with the same cycle shape. Automorphisms and the automorphism group Aut(G). The inner automorphism subgroup Inn(G). Inn(G) is isomorphic to G/Z(G). Example: Aut(S_n) = Inn(S_n) except when n = 6 (no proof).
• Lecture 7 (10/20/14): (Section 4.4, 4.5) If G is cyclic of order n then Aut(G) is isomorphic to (Z_n)^*. If G is the direct product of n copies of Z_p for a prime number p, then Aut(G) is isomorphic to GL_n(Z_p), the group of invertible n times n matrices with entries in Z_p. Introduction to the Sylow theorems and statement of the theorems.
• Lecture 8 (10/22/14): (Section 4.5) Proof of the Sylow theorems.
• Lecture 9 (10/24/14): (Section 4.5, 4.6) Proof that A_n is simple for n at least 5. Applications of Sylow theorems.
• Lecture 10 (10/27/14): (Section 4.5) Further applications of the Sylow theorems. Analysis of groups of order 105. Theorem on the recognition of an internal direct product.
• Lecture 11 (10/29/14): (Section 4.4) Definition of a semidirect product and the basic properties of semidirect products.
• Lecture 12 (10/31/14): (Section 4.4) Examples of semidirect products. Using semidirect products to classify groups of order pq. Groups of order 18.
• Lecture 13 (11/3/14): (Section 6.3) The free group. Presentations of groups. Examples.
• Lecture 14 (11/5/14): (Section 6.3, 6.1) Using a presentation of G to help find Aut(G). The commutator or derived subgroup [G, G] of a group G. Solvable groups. Examples: S_n is solvable only for n at most 4.
• Lecture 15 (11/7/14): (Section 6.1) The derived series of a group. A group is solvable if and only if the derived series reaches the bottom. Properties of solvable groups. Nilpotent groups. The upper central series Z_n(G) of a group G. A group is nilpotent if and only if the upper central series reaches the top.
• Lecture 16 (11/10/14): (Section 6.1) More on nilpotent groups. Normalizers group in nilpotent groups. A finite group is nilpotent if and only if all of its Sylow subgroups are normal. The lower central series: a group is nilpotent if and only if the lower central series reaches the bottom. Frattini's argument. A finite group is nilpotent if and only if all maximal subgroups are normal.
• Lecture 17 (11/12/14): (Section 7.1-7.3) Rings. Basic definitions and examples. Polynomial rings. Matrix rings.
• Midterm (11/14/14):
• Lecture 18 (11/17/14): (Section 7.4) Group rings. Basic ring technology: subrings, ideals, homomorphisms, fundamental homomorphism theorems. Prime and maximal ideals.
• Lecture 19 (11/19/14): (Section 7.4) A commutative ring R with only (0) and R as ideals is a field. An ideal I is maximal if and only if R/I is a field. Zorn's lemma. Proof that maximal ideals exist in a ring R (with 1, as always).
• Lecture 20 (11/21/14): (Section 7.5-7.6) The Chinese remainder theorem. Localization of a domain at a multiplicative system of nonzero elements. Special case: the field of fractions of an integral domain.
• Lecture 21 (11/24/14): (Section 8.1-8.2) Factorization in rings. Euclidean domains. Examples of Euclidean domains: Z, F[x] where F is a field, the Gaussian integers Z[i]. PIDs. A Euclidean domain is PID.
• Lecture 22 (11/26/14): (Section 8.2-8.3) Prime and inrreducible elements; associates. UFDs. PIDs are noetherian. In any domain primes are irreducible, and in a PID, irreducibles are prime. PIDs are UFDs.
• (11/28/14): NO CLASS (Thanksgiving)
• Lecture 23 (12/1/14): (Section 8.3) GCDs exist in PIDs, and gcd(a,b) = d is equivalent to (a, b) = (d). GCDs exist in UFDs. Examples of UFDs which aren't PIDs: R[x] where R is any domain which is not a field. Example of non-UFD: certain Quadratic integer rings (ex. Z[\sqrt{-5}]). Also the subring of F[x] consisting of polynomials with no x term.
• Lecture 24 (12/3/14): (Section 8.3) Factorization in Z[i]. Application to sums of two squares.
• Lecture 25 (12/5/14): (Section 9.3) If R is a UFD, then R[x] is a UFD. Main lemma due to Gauss: if f in R[x] factors as f = gh where g, h in F[x], where F is the field of fractions of R, then there are scalars a, b in F such that ag, bh in R[x] and f = (ag)(bh).
• Lecture 26 (12/8/14): (Chapter 9 ) Remainder and factor theorem. A polynomial f in F[x] is reducible if it has a root in F, and conversely if f has degree 2 or 3 then if f is reducible it has a root in F. Irreducible polynomials over the real and complex numbers. Rational root test. Eisenstein's criterion. If p is prime, then x^{p-1} + ... + x + 1 is irreducible in Q[x].

Homework