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)
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).
Class 5 March 11, 2025:
Discussion of homeworks 8-10. The second and third homomorphism theorems.
Commutator of a group, abelianization. Solvable groups.
Galois.
(Slides, pp.15-16,19-20,22-23; Lang, pp.16-17; Mac Lane-Birkhoff, pp.411,418;
Carter, p.183; Shafarevich, pp.154,156; Vinberg, pp.165-167,392).
Class 6 March 18, 2025:
Discussion of homeworks 11-15. Rings, fields, algebras. Examples: number fields,
matrices, polynomials, \(\mathbb Z / n\mathbb Z\) subject to addition and
multiplication modulo \(n\). Subrings, subalgebras. Homomorphisms, isomorphisms,
and automorphisms of rings and algebras. Rings consisting of 1 and 2 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 7 March 25, 2025:
Discussion of homeworks 16-19. The fields GF(2) and GF(3). Ideals and quotiens,
examples. Principal ideals of a commutative ring. Simple rings and algebras. The
matrix algebra \(M_n(K)\) is simple. The first, second and third homomorphism
theorems for rings and algebras. Quaternions.
(Slides, pp.29-30,44; Lang, pp.86-89; Mac Lane-Birkhoff, pp.95-98,281-283;
Shafarevich, pp.26,28-29,65-66; Vinberg, pp.459-460).
Class 8 April 1, 2025:
Discussion of homeworks 19-21. Any finite-dimensional algebra is isomorphic to a
subalgebra of a matrix algebra. Examples showing that the relation
"to be an ideal" in rings is not transitive. Algebra over an extension of the
base field. Quaternion algebra over \(\mathbb C\) is isomorphic to
\(M_2(\mathbb C)\). Tensor product of algebras. Isomorphisms
\(K[x] \otimes K[y] \simeq K[x,y]\), \(M_n(A) \simeq M_n(K) \otimes A\).
(Slides, p.48-50; Mac Lane-Birkhoff, pp.319-325; Vinberg, pp.460-461).
Class 9 April 8, 2025:
Discussion of homeworks 22-23. Characteristic of a field. Prime subfields.
Finite fields. Construction of GF(4). Algebraic and transendental extensions of
fields.
(Lang, pp.223-225,244-247; Mac Lane-Birkhoff, pp.120-121).
Class 10 April 22, 2025:
Algebraically closed fields, algebraic closure. The fundamental theorem of
algebra. Passing from an algebra to the algebra over the algebraic closure of
the ground field. Nilpotent elements, nilpotent algebras.
(Slides, p.36; Vinberg, pp.93-97,359,434).
Class 11 April 29, 2025:
Radical, semisimple algebras, trace form.
(Vinberg, pp.434-439).
Class 12 May 6, 2025:
Lie algebras: examples, center, commutant, classification of algebras of low
dimension, derivation algebra of an algebra, relationship between derivation and
automorphism via exponentiation.
(Shafarevich, pp.189-192).
(referred below as "Slides")