Lecture plan
Date | Covered |
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9/24 | 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. |
9/26 | Notation for elements of D_{2n}. Notation for elements of Symmetric groups: Cycles and cycle decomposition. Proof that any element can be written as a product of disjoint cycles (we'll prove uniqueness later). The matrix group GL_{n}(F) where F is a field. Read on your own: The quaternion group Q_{8}. |
9/28 | Homomorphism, Isomorphism. Subgroups. Cyclic groups and cyclic subgroups. |
10/1 | More on cyclic groups and cyclic subgroups. Two cyclic groups are isomorphic if and only if they have the same number of elements (finite or infinite). The order of x^{a} in terms of the order of x. Classification of subgroups of a cyclic group G: For finite order n, they're in bijection with the divisors of n, for infinite G they're in bijection with the non-negative integers. |
10/3 | Briefly talked about the lattice of subgroups and how to depict it. Generators for subgroups generated by more than one element: the smallest subgroup containing the generators, defined as the intersection of all subgroups containing the generators (the proof that this is a group is in the book, but we skipped it in class). Then we carefully analyzed the question of how to find a surjective homomorphism out of a given group G with a specified kernel K: First we proved that such a thing is unique up to unique isomorphism if it exists. Secondly we proved that it's necessary that K be normal (in fact we proved that the kernel of any homomorphism is normal). Finally, we defined the left cosets gH of a subgroup H of G and the set G/H of left cosets. (The book only uses the notation G/H in the case where H is normal, but it's common to use it in the general case too.) Next time we'll see that if H is normal, then there's a natural group structure on G/H. |
10/5 | More on quotient groups. Proof that if H is normal in G, then there is a well defined group structure on G/H given by (g_{1}H)(g_{2}H) = (g_{1}g_{2})H. Examples of a subgroup of D_{6} which is normal and one which is not normal. Lagrange's theorem. |
10/8 | Multiple consequences of Lagrange's theorem. The normalizer of a subgroup. Centralizer and center. Examples. |
10/10 | Subgroups of index 2 are normal. First isomorphism theorem. Simple groups. Mentioned the classification of finite simple groups. |
10/12 | Symmetric groups are generated by 2-cycles. The sign homomorphism. The alternating group A_{n}. The alternating group is generated by 3-cycles. A_{n} is simple when n is at least 5 (proof given next time). |
10/15 | A_{n} is simple for n at least 5. Solvable groups. S_{n} is solvable if and only if n is at most 4. |
10/17 | Groups acting on sets: axioms and examples. Orbits, stabilizers, and their relation (often called the "orbit-stabilizer theorem", although the book doesn't use that phrase. The content is a bijection between the orbit of an element and the set of left cosets of the stabilizer of that element). Applications: Cauchy's theorem (I gave a proof which in the book is essentially exercise 3.2.9) and Cayley's theorem. |
10/19 | Symmetry groups of the platonic solids. The action of a group G on the set G/H of left cosets of a subgroup H. A subgroup is normal if its index is the smallest prime number dividing the order of G. The conjugation action of G on itself and the class equation. Application: p-groups have non-trivial center. |
10/22 | Sylow p-subgroups. Definitions, examples, and beginning of proof. |
10/24 | Midterm exam. |
10/26 | Finished proof of Sylow's theorems. Examples: Groups of order pq. Simple groups of order 60 are isomorphic to A_{5}. |
10/29 | Simple groups of order 60 (finished proof). Recognition of direct products. The WIM assignment is related. |
10/31 | Beginnings of ring theory. Axioms and examples. |
11/2 | Ring theory. Subring, Ideals, quotient rings, first isomorphism theorem for rings. Polynomial rings. Generators for ideals |
11/5 | The Chinese remainder theorem. Lecture taught by Akshay Venkatesh. |
11/7 | More about rings and ideals: Integral domains. Maximal and prime ideals in a commutative ring with 1. Examples. |
11/9 | Prime and irreducible elements in an integral domain. Prime implies irreducible. Unique Factorization Domains. Principal Ideal Domains. Example: Z[sqrt(-5)] is not a UFD. |
11/12 | Associated elements. PID implies UFD (this is the main result of this section.) Euclidean domains. |
11/14 | Euclidean domains are PID's (and hence UFD's). Examples of Euclidean domains: the integers Z, polynomial rings F[x] over a field F. Discussed classification of irreducible elements in F[x], for various fields F (for the complex numbers we proved in class that a polynomial is irreducible if and only if it's degree one, for the real numbers we stated without proof that the irreducible elements are the polynomials of degree one, and the degree two polynomials with negative discriminant, for rational numbers and finite fields the question is very interesting, but difficult.) |
11/16 | More on irreducible elements in polynomial rings. Polynomials of degree 2 and 3 are irreducible if and only if they have no roots. For example, the polynomial f(x) = x^{2} + x + 1 over the field with two elements is irreducible. This is implies that (f) is maximal; the quotient ring is a field with four elements. Gaussian integers Z[i]: they form a euclidean domain and hence a PID and hence a UFD. Aside on Lame's false proof of FLT. |
11/26 | Fermat's theorem on primes which are sums of two squares, and its relation to Gaussian primes. |
11/28 | The characteristic of a ring with identity. Any finite subgroup of the group of units in a field is cyclic. Finite fields: The characteristic is a prime number, and the number of elements is a power of that prime number. |
11/30 | Existence of finite fields with p^{n} elements. Multiple ingredients in the construction: 1. Any polynomial over any field splits as a product of linear factors, after possible enlarging the field. 2. The Frobenius homomorphism of a commutative ring of characteristic p. 3. A field with p^{n} elements was then constructed as the set of roots of the polynomial f(x) = x^{pn} - x, in a field where that polynomial factors as the product of linear factors. 4. Differentiation of polynomials was used to prove f(x) had not repeated roots. 5. As a corollary, there exist irreducible polynomials over F_{p} of any degree. |
12/3 | Uniqueness of finite fields: First we proved that any finite field K with p^{n} elements is isomorphic to F_{p}[x]/(f) for some irredicible polynomial f(x) of degree n which is a factor of x^{pn}-x. Then we proved that in K[x], the polynomial x^{pn} - x factors as the product of x - t over all t in K. Then we used that K[x] is a UFD to see that any f(x) which is a factor of x^{pn} - x has a root over K. Finally, if K_{1} is a field with p^{n} elements, isomorphic to F_{p}[x]/(f), we get an isomorphism to another field K_{2} with p^{n} elements by sending g + (f) to g(t), where t is an element of K_{2} with f(t) = 0. At the end of the lecture, we briefly discussed when F_{pn} contains a subfield isomorphic to F_{pm}. This happens if and only if m divides n. |
12/5 | Review of group theory. |
12/7 | Review of ring theory. |