Tuesday 11:00-approx.14:00 (depending on the breaks) G401
SYLLABUS
Algebraic structures with one binary operation.
Groups, subgroups. Abelian groups.
Homomorphisms, isomorphisms, automorphisms.
Cyclic groups.
Decomposition of a group with respect to a subgroup. Normal subgroups,
simple groups.
Quotients. Direct products. Homomorphism theorems.
Algebraic structures with two binary operations.
Ideals, quotient rings.
Tensor products.
Commutative rings, rings of polynomials, rings of matrices, quaternions.
Fields. Prime field.
Algebraically closed fields.
Finite fields.
LITERATURE
(see here for the precise titles and further info)
Main:
Lang
Mac Lane-Birkhoff
Additional:
Carter
Shafarevich
Vinberg
A brief presentation covering some of the topics:
(referred below as "Slides")
Group Explorer
A BRIEF SYNOPSIS
(each class lasted approximately 3 hours, unless specified otherwise)
Class 1 February 11, 2025:
Organizational issues. General notion of an algebraic system. The idea of
symmetries. Symmetries of an equilateral triangle form the group \(S_3\).
Definition of a group, examples of groups (permutations, matrices, residues
from the division by \(n\)). Subgroup. Abelian groups.
Abel.
The inverse of an element in a group is unique. Groups of order 1,2,3.
(Slides, pp.3-9; Lang, pp.7-9; Mac Lane-Birkhoff, pp.43-48,50,63-64;
Carter, pp.3-21,25-40,45,48-51,78-80; Shafarevich, pp.96-102,108,110;
Vinberg, pp.139-140).
Class 2 February 18, 2025:
Subgroup generated by a subset. Homomorphism, isomorphism, automorphism of
groups, examples. Automorphism group. Inner automorphisms. Cyclic groups.
Order of an element of a group. Direct product of groups (defintion).
(Slides, pp.10-11,17-18; Lang, pp.8-10,26; Mac Lane-Birkhoff, pp.43-44,46;
Carter, pp.117-121,159-163; Shafarevich, pp.104-105; Vinberg, pp.163-165).
Class 3 February 25, 2025:
Discussion of homeworks 1-3. Two non-isomorphic groups of order 4. Center of a
group, examples. The special linear group. Properties of the direct product:
center, order of elements. Cosets, decomposition of a group with respect to
subgroup. Lagrange's theorem. Normal subgroups, examples.
(Slides, p.12; Lang, pp.13-14; Mac Lane-Birkhoff, pp.72-74; Carter, pp.102-108;
Shafarevich, pp.105-106,109; Vinberg, pp.155-158,161).
Class 4 March 4, 2025:
Discussion of homeworks 4-7. The center is a normal subgroup, inner automorhisms
form a normal subgroup in the group of automorphisms. Quotients. Kernel of a
homomorphism. The first homomorphism theorem. Simple groups.
(Slides, pp.12-15,24; Lang, pp.11-14,16; Mac Lane-Birkhoff, pp.pp.75-77,79-80;
Carter, pp.132-139,163-169; Shafarevich, pp.106-107,109;
Vinberg, pp.161-162,165-168,404-405).
(referred below as "Slides")