Decomposition of groups with respect to subgroups. 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:
Group Explorer
A BRIEF SYNOPSIS
(each class lasted approximately 3 hours, inless specified otherwise)
Class 1 February 12, 2024:
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). Subgroup. Abelian groups.
The inverse of an element in a group is unique, the unit in a group is unique.
Groups of order 2.
Galois and
Abel.
(Slides, pp.3-8; 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 19, 2024:
Cyclic groups. Order of an element of a group. Groups of order 3. Homomorphism,
isomorphism of groups, examples. Example of two non-isomorphic groups of order
4.
(Slides, pp.7,10-11,18; Lang, pp.8-10; Mac Lane-Birkhoff, p.44;
Carter, p.159-163; Shafarevich, pp.104-105; Vinberg, pp.163-165).
Class 3 February 26, 2024:
Discussion of homeworks 1-2. Center of a group, examples. The special linear
group. Direct product of groups, center of the direct product.
(Slides, p.17; Lang, p.9; Carter, pp.117-128).
Class 4 March 4, 2024:
Discussion of homeworks 3-6. Automorphism group, examples. Inner automorphisms.
Normal subgroup.
(Slides, p.11-12; Lang, pp.13-14,26; Carter, pp.132-134;
Shafarevich, pp.105-106; Vinberg, pp.161-162).
Class 5 March 18, 2024:
Discussion of homeworks 7-8. Cosets. Lagrange's theorem and its consequences.
Quotients. Simple groups.
(Slides, pp.12-14,24; Lang, pp.12-13,26; Mac Lane-Birkhoff, pp.72-74,79;
Carter, pp.102-108,132-139,163-167; Shafarevich, pp.105-106,109;
Vinberg, pp.155-158,161-162).
Class 6 March 25, 2024:
Discussion of homeworks 9-11. The group of inner automorphisms is a normal
subgroup of the group of all automorphisms. Kernel.
The first, second and third homomorphism theorems.
(Slides, pp.15-16; Lang, pp.11,16-17; Mac Lane-Birkhoff, pp.75-77,79,411;
Carter, pp.163-169; Shafarevich, pp.106-107; Vinberg, pp.165-167).
Class 7 April 8, 2024:
Discussion of homeworks 12-13. Rings, fields, algebras. Examples: number fields,
matrices, polynomials, GF(2), GF(3). Homomorphism and isomorphism of rings.
Rings consisting of 1,2,3 elements, algebras of dimension 1.
(Slides, pp.26-27,33-35,41-42; Lang, pp.83-84,86; Mac Lane-Birkhoff, pp.85-87;
Shafarevich, pp.17-18,62-63; Vinberg, pp.7-9,27-32;
note that both Lang and Mac Lane-Birkhoff assume that rings have a unit).
Additional read: B. Poonen,
Why all rings should have a 1,
Math. Magazine 92 (2019), no.1, 58-62.
Class 8 April 15, 2024 (1h 20min):
Discussion of homeworks 14-17.
Class 9 April 22, 2024:
Discussion of homeworks 18-19. Subrings and subalgebras. Ideals and quotiens,
examples. Principal ideals of a commutative ring. Simple rings and algebras. The
matrix algebra \(M_n(K)\) is simple. The first isomorphism theorem.
(Slides, pp.29-30, Lang, pp.86-89; Mac Lane-Birkhoff, pp.95-98;
Shafarevich, pp.26,28-29; Vinberg, pp.11-12)
Class 10 April 29, 2024:
Discussion of homework 20. The second and third homomorphism theorems for rings
and algebras. Quaternions. Tensor product of algebras.
Simplicity of the quaternion algebra:
(Slides, p.44,48-50; Mac Lane-Birkhoff, pp.281-283,319-325;
Shafarevich, pp.65-66; Vinberg, pp.295-300,459-460).
Class 11 May 6, 2024 (2h30min):
Characteristic of a field. Prime subfields. Finite fields. Construction of
GF(4). Algebraic and transendental extensions of fields. Algebraically closed
fields, algebraic closure.
(Lang, pp.223-224,244-247; Mac Lane-Birkhoff, pp.120-121,281-283;
Shafarevich, pp.65-66; Vinberg, pp.459-460).
