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