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 1.5 hours)
Class 1 February 12, 2026:
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, \(\mathbb Z / n\mathbb Z\)). Abelian groups. The
inverse of an element in a group is unique.
Galois and
Abel.
(Slides, pp.3-8; Lang, pp.7-10; Mac Lane-Birkhoff, pp.43-48,50,63-64;
Carter, pp.3-21,25-40,45,48-51,78-80; Shafarevich, pp.96-101,108,110;
Vinberg, pp.139-140).
Class 2 February 19, 2026:
The general linear group. Homomorphism, isomorphism of groups. Examples: parity
of a permutation, determinant as group homomorphisms. Classification of groups
of order 1 and 2.
(Slides, pp.10-11; Lang, p.10; Shafarevich, pp.104-105; Vinberg, pp.163-165).
Class 3 February 26, 2026:
Subgroup, examples of subgroups. Subgroup generated by a subset. Cyclic groups.
Classification of groups of order 3.
(Slides, pp.9,18; Lang, p.9; Mac Lane-Birkhoff, pp.51-54; Shafarevich, p.104).
Class 4 March 5, 2026:
Direct product of groups. Order of an element in a group, order of elements in
direct products. Classification of groups of order 4. Ways to establish
non-isomorphism of groups. Automorphism group of a group.
(Slides, pp.10,17; Lang, pp.9,26; Mac Lane-Birkhoff, p.46; Carter, pp.117-121)
Class 5 March 19, 2026:
Examples of computation of automorphism group: cyclic groups of small order,
\(S_3\). Inner autmorphisms. Center of a group, examples.
(Slides, p.11; Lang, p.26).
Class 6 March 26, 2026:
Cosets, decomposition of a group with respect to a subgroup. Lagrange's theorem.
Normal subgroups, examples. Quotient group.
(Slides, pp.12-14,24; Lang, pp.12-14; Mac Lane-Birkhoff, pp.72-74,79-80;
Carter, pp.102-108,132-139; Shafarevich, pp.105-107;
Vinberg, pp.155-158,161-162).
Class 7 April 9, 2026:
Simple groups. \(S_n\) is not simple. Abelian simple groups are exactly
\(\mathbb Z / p\mathbb Z\). Kernel of a homomorphism, examples. Kernels and
normal subgroups are, essentially, the same things. The first and the second
homomorphism theorems.
(Slides, pp.15,24; Lang, pp.11,14,16-17; Mac Lane-Birkhoff, pp.75-77,79-80,411;
Carter, pp.163-169; Shafarevich, pp.106-107,109,154-155;
Vinberg, pp.162,164-168,404-405).
Class 8 April 16, 2026:
The third homomorphism theorem for groups. Rings and fields: definition,
examples.
(Slides, pp.16,26-27,33-35; Lang, pp.17,83-86; Mac Lane-Birkhoff, pp.85-87,411;
Shafarevich, pp.17-18,62-63,154; Vinberg, pp.7-9;
note that both Lang and Mac Lane-Birkhoff assume that rings have a unit).
Class 9 April 30, 2026 (3h):
Algebras. Subrings, subalgebras. Homomorphisms, isomorphisms, and automorphisms
of rings and algebras. Rings consisting of 1 and 2 elements. Ideals and
quotiens, examples. Principal ideals of a commutative ring. Simple rings and
algebras. The matrix algebra \(M_n(K)\) is simple (only idea of the proof,
without details).
(Slides, pp.28-30,41-42; Lang, pp.86-88; Mac Lane-Birkhoff, pp.87,95-97;
Shafarevich, pp.26,28; Vinberg, pp.27-32).
Class 10 May 7, 2026 (3h):
Kernel of the homomorphism. The first, second and third homomorphism theorems
for rings and algebras. Direct sum of rings. Extension of fields.
Algebraically closed fields. The fundamental theorem of algebra (without proof).
Characteristic of a field. Prime subfields. Finite fields. Construction of
GF(4).
(Slides, pp.29-31,35-39,44; Lang, pp.88-90,223-224,244-246;
Mac Lane-Birkhoff, pp.95-98,120-121; Shafarevich, pp.26,28-31; Vinberg, p.93).
Additional read: B. Poonen,
Why all rings should have a 1,
Math. Magazine 92 (2019), no.1, 58-62.
(referred below as "Slides")