Week 1, Septermber 9, 2009 9/09: Peano axioms, mathematical induction. Proof by induction: the existence of the division algorithm, by induction. associativity of addition and multiplication. Modular arithmetics. 9/11: Definition of groups, rings, fields. examples: the ring of integers, rational numbers, real numbers complex numbers, matrix rings, matrix groups 9/14: Subgroups, subrings, sufields, examples. equivalence relations, equivalence classes modular arithmetic (Z/nZ as finite rings), congruence, equivalence relation, the natural homomorphism from Z to Z/nZ, 9/16: modular arithmetic continued, proof that addition and multiplication on Z/nZ is well defined, definition of integral domains, integral domains among all Z/nZ's, Z/pZ is a field (when p is a prime number)--as a consequence of being a finite integral domain, product rings and product groups. 9/18: more on product rings and product groups, application to structure of units of Z/nZ, division algorithm, determine all subgroups of Z, definition of dihedral groups (as symmetries of a regular n-gon) definition of Ad_G as a homomorphism from G to Aut(G) 9/21: dihedral groups continued, structure of dihedral groups, finite general linear groups, definition of the center of a group, definition of the centralizers of a subset, definition of the normalizer of a subgroup 9/23: symmetric groups/permutation groups, examples, Klein four group in S_4, Hamiltonian quaternions, quaternion groups 9/25: left and right cosets, definition of [G:H], order of an alement divides the order of the group, Fermat's little theorem, conjugation, normal subgroups conjugacy classes, the conjugacy class of an element x is in bijection with cosets w.r.t. the centralizer of x. 9/28: Review: Fermat's little theorem, class equation. Application: every non-trivial p-group has a non-trivial center. Examaples of groups with trivial center. Recall the definition of normal subgroups. The kernel of a ring homomorphism is a normal subgroup. 9/30: Homomorphisms from a cyclic group to an arbitrary group. Review: the kernel of a group homomorphism is a normal subgroup. Construction of the quotient group and the canonical homomorphism. 10/2: Review of construction of quotient groups. Examples of quotients: Z/nZ, R/Z, C/Z. Universal property of the quotient homomorphism. 10/5: The image of a homomorphism is isomorphic to the quotient by the kernel. Characterization of the quotient homomorphisms, correspondence between subgroups of a quotient group and subgroups in the target group, isomorphism theorems in the quotient situatiion. 10/7 Definition of ideals. The kernel of a homomorphism of rings is an ideal. Construction of quotient rings. Analog for rings. Definition of vector spaces and linear transformations. 10/9: Linear combinations, linear span, linear independence, Use of basis (= putting a linear coordinate system on a vector space) Statement of the main result for the dimension of a vector space (chap 3, sec 3 of Artin) 10/12: Proof that any two basis of a vector space have the same number of elements. Equivalence of alternative definition of basis. (chap 3, sec 3 of Artin) 10/14: Review of quotient vecor spaces and their dimension. Examples of linear tranformations and matrix representations. (matrix respresentation is discussed in chap 4, sec 2 of Artin) Change of basis (discussed in chap 4, sec 4 of Artin) 10/16: Change of basis and effect on matrix representation (chap 3 sec 4 and chap 4 sec 2 of Artin). Review: eigenvalues and eigenvectors (chap 4, sec 3 and sec 4 of Artin) 10/23: Change of basis continued. Examples. 10/26: Eigenvalues, eigen vectors, expansion formula for determinants, sign of a permutation. 10/28: More about determinants, 10/30: diagonlization: diagonaliazable linear operators. Examples. 11/02: Definition of linear representation of a group G (it is a homomorphism from G to a GL(n)). Examples for dihedral group, rotation group, symmetric group and the general linear group. Definition of group action (chap 5, sec 5 of Artin) Note: Chap 5, sec 1-4 of Artin contains many concrete examples of group actions. 11/04: Definition of orbits, stabilizer subgroup, fixer subgroup (chap 5 sec 5 of Artin). Examples: D_{2n} operating on the plane, adjoint representation Parametrization of a G-orbit by G/Stab_G(x), (chap 5, sec 6 of Artiin) counting "forumla" (chap 5, sec 7) 11/06: Adjoint action as example of group action (review). Translation action on G/H. Revisit: a non-trival p-group has a non-trivial center. Parametrize all subgroups conjugate to a given subgroup. Permutation representation. Every element of S_n is a product of disjoint cycles. 11/09: Sylow theorems. See Chap 6, sec 4 of Artin. Proof of the existence part of Sylow theorems. (Wieland's proof) 11/11: Proof of the second and the third part of Sylow's theorem (using counting arguments involving group action). 11/13: Example: classification of all groups with 15 elements. Revisit: characterization of product groups. Beginning of another proof of the existence part of Sylow's theorem. 11/16: Continuation: another proof of the existence part of Sylow's theorem, by induction. 11/18: Application of Sylow's theorem. Every group with 15 elements is cyclic. Every group with 12 elements either has a normal Sylow 2-subgroup or a normal Sylow 3-subgroup. 11/20: Every group with 12 elements and a normal Sylow 2-subgroup is either commutative or isomorphic to the alternatiing group on 4 letters. 11/23: Semi-direct product: definition, application to classification of groups with 12 elements. 11/25: minimal polynomial of a linear operator. Use of the polynomial ring in linear algebra. polynomial rings over fields. Every ideal of Z is principal (i.e. generated by one element). Every ideal of a polynomial ring over a field is principal. 11/30 How to use the polynomial ring for studying a linear operator. 12/02 primary decompositon into direct sum of generalized eigenspaces of a linear operator 12/04 factorization in a principal ideal domain