6ALGS Algebraic Structures, summer semester 2022/2023

A BRIEF SYNOPSIS
(each class lasted approximately 1.5 hours, lectures only)

Class 1 February 17, 2023:
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 (permutation, matrices). Abelian groups. 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 24, 2023:
The inverse of an element in a group is unique. Cyclic groups. Order of an element of a group. Groups of order 3. Subgroup. Alternating groups. (Slides, pp.7,9; Lang, pp.8-10; Mac Lane-Birkhoff, pp.44,47-49).

Class 3 March 3, 2023:
Subgroups of \(S_3\), \(\mathbb Z\), the general linear group. Homomorphism, isomorphism of groups, examples. Example of two non-isomorphic groups of order 4. Center of a group. How to decide whether groups are isomorphic or not. (Slides, p.11; Lang, pp.10-11; Shafarevich, pp.104-105; Carter, p.159-163; Vinberg, pp.163-165).

Class 4 March 10, 2023:
Automorphism group. Inner automorphisms. Cosets. Lagrange's theorem and its consequences. (Slides, p.11; Lang, pp.12-13,26; Mac Lane-Birkhoff, pp.72-74; Carter, pp.102-108; Shafarevich, pp.105-106; Vinberg, pp.155-157).

Class 5 March 17, 2023:
Example when right cosets and left cosets are not the same (\(S_3\)). Normal subgroups. Quotients. (Slides, p.12-14; Lang, pp.13-14; Carter, pp.132-139; Vinberg, pp.161-162).

Class 6 March 24, 2023:
Simple groups. Kernel. The first and third (!) homomorphism theorems. (Slides, p.15-16,24; Lang, p.11,16-17; Mac Lane-Birkhoff, p.75-77,79,411; Carter, p.163-169; Shafarevich, p.106-107,109; Vinberg, pp.165-166).

Class 7 March 31, 2023:
Examples of application of the first homomorphism theorem (cyclic groups, \(GL_n\), \(SL_n\)). The second homomorphism theorem. Direct product of groups. (Slides, pp.14-15,17; Lang, p.17; Mac Lane-Birkhoff, p.411; Carter, p.117-128).

Class 8 April 14, 2023:
Definitions and examples of rings and algebras. Classification of rings consisting of two elements. Subrings and subalgebras. (Slides, pp.26-27,41-42; Lang, pp.83-84; Mac Lane-Birkhoff, pp.85-87,90; Shafarevich, pp.17,62; Vinberg, pp.7-12,27,30-32; note that both Lang and Mac Lane-Birkhoff assume that rings have a unit).

Class 9 April 21, 2023:
Fields. Homomorphisms, automorphisms, automorphism group of a ring (algebra). Ideals, examples. Principal ideals of a commutative ring. Simple rings. The matrix ring over a field is simple. (Slides, pp.29,33-35; Lang, pp.86-87; Shafarevich, p.26; Vinberg, pp.9,12).

Class 10 April 28, 2023:
Quotients, examples. Theorem: any field is simple as a ring. Ideals are "the same things" as kernels of homomorphisms. Structure constant and multiplication table of an algebra. (Slides, pp.30,34,43; Lang, p.89; Mac Lane-Birkhoff, pp.95-98,100; Shafarevich, p.26; Vinberg, pp.28-29).

Class 11 May 5, 2023:
First, second, and third homomorphism theorems for rings. Direct sum of rings. Tensor product of algebras. Isomorphism \(M_n(A) \simeq M_n(K) \otimes A\). (Slides, p.31,48-50; Mac Lane-Birkhoff, pp.319-325; Shafarevich, pp.28-29; Vinberg, pp.295-300).

Class 12 May 16, 2023:
Quaternions. Characteristic of a field. Prime subfields. Finite fields. Construction of \(GF(4)\). (Lang, pp.244-247; Mac Lane-Birkhoff, pp.120-121,281-283; Shafarevich, pp.65-66; Vinberg, pp.459-460).


6ALGS Algebraic Structures, summer semester 2021/2022

A BRIEF SYNOPSIS
(each class lasted approximately 3 hours)

Class 1 February 14, 2022: General notion of algebraic system. The idea of symmetries. Symmetries of the equilateral triangle form the group \(S_3\). Definition of the group, examples of groups. Subgroup. Abelian group. Symmetric group, calculations with permutations. Group consisting of 1 element is trivial, group consisting of 2 elements is isomorphic to \(\mathbb Z / 2\mathbb Z\). Group consisting of 3 elements is isomorphic to \(\mathbb Z / 3\mathbb Z\). Order of a group, order of an element in a group. Cyclic groups. (Slides, pp.3-9,18,20; Lang, pp.7-9; Mac Lane-Birkhoff, pp.43-48,51-53,56,63-64; Carter, pp.3-15,25-40,45,48-51,78-80; Shafarevich, pp.96-102,108,110,151-152; Vinberg, pp.139-141,147).

Class 2 February 21, 2022: Subgroups of small groups (cyclic, \(S_3\)). Homomorphism, isomorphism, examples. Center of a group. How to decide whether groups are isomorphic or not. (Slides, p.11; Lang, pp.10-11; Shafarevich, pp.104-105; Carter, p.159-163; Vinberg, pp.163-165).

Class 3 February 28, 2022: Classification of cyclic groups. Automorphism group. Inner automorphisms. Cosets. Lagrange's theorem and its consequences. Normal subgroups. (Slides, p.12,18; Lang, pp.13-14,23-25; Mac Lane-Birkhoff, pp.51-54,72-74,77; Carter, pp.102-108; Shafarevich, pp.105-106; Vinberg, pp.152,155-157,161).

Class 4 March 7, 2022: Quotients. Simple groups. Kernel. The first homomorphism theorem. (Slides, p.12-15,24; Lang, p.11,13-14,16; Mac Lane-Birkhoff, p.75-77,79; Carter, p.132-139,163-169; Shafarevich, p.106-107,109; Vinberg, pp.161-162,164-166).

Class 5 March 14, 2022: Discussion of homeworks 1-5. Direct products. The second homomorphism theorem. (Slides, p.15; Lang, p.17; Mac Lane-Birkhoff, p.411; Carter, pp.117-128).

Class 6 March 21, 2022: Discussion of homework 6. Commutant of a group. The third homomorphism theorem. Solvable and nilpotent groups, Abel and Galois (sketchy, without any details). (Slides, pp.16,19-23; Lang, p.17; Mac Lane-Birkhoff, p.411,418; Carter, p.183; Shafarevich, pp.154,156; Vinberg, p.392).
Material for self-study: \([S_n,S_n] = A_n\), \([GL_n(K),GL_n(K)] = SL_n(K)\) (Vinberg, pp.392-393).

Class 7 March 28, 2022: Definitions and examples of rings, algebras, fields. Classification of rings consisting of two elements. Subrings and ideals. Homomorphism of rings, automorphism group of rings. Theorem: any field is simple as a ring. (Slides, pp.26-29,33-34,41-42; Lang, pp.83-84,86-88; Mac Lane-Birkhoff, pp.85-87,90,99-100; Shafarevich, pp.17,26,62; Vinberg, pp.7-12,27,30-32; note that both Lang and Mac Lane-Birkhoff assume that rings have a unit).

Class 8 April 4, 2022: Discussion of homeworks 7-10. Quotients of rings and algebras. Simple rings. \(M_n(K)\) is simple. Examples showing that the relation to be an ideal is not transitive. Kernels. The first homomorphism theorem for rings. (Slides, pp.29-30; Mac Lane-Birkhoff, pp.95-98; Shafarevich, pp.28-29).

Class 9 April 11, 2022: The second and third homomorphism theorems for rings. Direct product of rings. Tensor product of algebras. (Slides, p.31,48-50; Mac Lane-Birkhoff, pp.319-325; Vinberg, pp.295-300).

Class 10 April 25, 2022: Discussion of homeworks 11-15. Isomorphisms \(K[x] \otimes K[y] \simeq K[x,y]\) and \(M_n(A) \simeq M_n(K) \otimes A\).

Class 11 May 2, 2022: Quaternions. Quaternions over \(\mathbb R\) form a division algebra, quaternions over \(\mathbb C\) are isomorphic to the \(2 \times 2\) matrix algebra. Quaternion group \(Q_8\). Algebraically closed fields, algebraic closure. Isomorphism \(\mathbb R[x]/(x^2 +1) \simeq \mathbb C\). Evaluation maps. (Mac Lane-Birkhoff, pp.281-283; Shafarevich, pp.65-66; Vinberg, pp.459-460).

Class 12 May 10, 2022: Discussion of homeworks 16-18. Characteristic of a field. Prime subfields. Finite fields. Algebraic and transendental extensions of fields. Construction of \(GF(4)\). (Lang, pp.223-224,244-247; Mac Lane-Birkhoff, pp.120-121).
Simplicity of the quaternion algebra.

HOMEWORKS


6/7ALGS Algebraic Structures, summer semester 2020/2021

A BRIEF SYNOPSIS

Class 1 February 24, 2021 (approx. 2 hours): General notion of algebraic system. Algebraic systems with one binary operation. Examples. Multiplication table. Definition of group, examples. Definition of semigroup. Group of symmetries of pentagon is isomorphic to \(\mathbb Z / 5\mathbb Z\). Symmetric group, calculations with permutations. Group consisting of 1 element is trivial, group consisting of 2 elements is isomorphic to \(\mathbb Z / 2\mathbb Z\). (Slides, pp.3-8; Lang, pp.3-4,6-8; Mac Lane-Birkhoff, pp.39-40,43-48,63-64; Carter, pp.3-15,25-40,45,48-51,78-80; Shafarevich, pp.96-102,108,110,151-152; Vinberg, pp.139-140).

Class 2 March 3, 2021 (approx. 2 hours 10 min): Group consisting of 3 elements is isomorphic to \(\mathbb Z / 3\mathbb Z\). The inverse of any element is unique. Cancellation from the left and from the right. Definition of powers of elements in a group. Subgroups, examples of subgroups. Alternating group. Abelian groups. Abel and Galois. Homomorphisms, isomorphisms and automorphisms, examples. Inner automorphism. (Slides, pp.7,9-11; Lang, pp.9-10; Mac Lane-Birkhoff, 43-44,50,56-57,67; Shafarevich, pp.104-105; Vinberg, pp.141-143).
video

Class 3 March 10, 2021 (approx. 2 hours 15 min): Left and right cosets, Lagrange's theorem, normal subgroup, quotient of a group by a normal subgroup, simple groups. Examples: abelian groups, \(\mathbb Z / n\mathbb Z\), \(A_n \triangleleft S_n\), \(SL_n(K) \triangleleft GL_n(K)\). (Slides, pp.12-14,24; Lang, pp.12-14; Mac Lane-Birkhoff, pp.72-73,79-80; Carter, pp.102-108,169-172; Shafarevich, pp.105-106,159-160; Vinberg, pp.155-157,161-163).
video

Class 4 March 17, 2021 (approx. 2 hours 10 min): Normal subgroups are kernels of homomorphisms. First, second and third homomorphism theorems. Applications of the first homomorphism theorem. The order of an element in a group. Cyclic groups. (Slides, pp.13,15-16,18; Lang, pp.23-24; Mac Lane-Birkhoff, pp.51-54,77; Carter, pp.64-66; Shafarevich, pp.107,154; Vinberg, pp.147-153).
video
Material for self-study: An example showing that the normality in groups is not transitive.

Class 5 March 24, 2021: Rings. Rings consisting of two elements are: GF(2) and the square-zero ring. Examples of rings. (Slides, pp.26-27; Lang, pp.83-86; Mac Lane-Birkhoff, pp.85-87; Shafarevich, pp.17-19,62; Vinberg, pp.7-9).
video   notes (incomplete)

Class 6 March 31, 2021: Homorphisms of rings, examples and non-examples (trace and determinant). Subrings, ideals, quotient rings. Examples. (Slides, pp.28-30; Lang, pp.86-88; Mac Lane-Birkhoff, pp.87,95-97; Shafarevich, pp.24-25,28,68-69; Vinberg, pp.337-339).
video

Class 7 April 7, 2021: Ideals and ring homomorphisms (continuation). First homomorphism theorem. 2nd and 3rd homomorphism theorems (briefly, without proofs). (Slides, p.29; Lang, p.89; Mac Lane-Birkhoff, pp.96-97; Shafarevich, pp.28-29; Vinberg, pp.339,341).
video   notes

Class 8 April 14, 2021: Examples showing that the property to be an ideal is not transitive. Simple rings, simplicity of the full matrix ring. Direct products of groups and direct sums of rings.
video   notes

Class 9 April 21, 2021: Direct products of groups and direct sums of rings (continuation). Commutative rings. Algebras. Algebras of polynomials and matrices. (Slides, p.31,41-42; Lang, pp.97-98,503-504; Mac Lane-Birkhoff, pp.104-106,225-227; Shafarevich, pp.17-20; Vinberg, pp.27-33,81-83).
video   notes (incomplete)

Class 10 April 28, 2021: Quaternions. Isomorphism of quaternions over complex numbers with the 2x2 matrix algebra. Fields. (Slides, pp.33-34,44; Mac Lane-Birkhoff, pp.281-283; Shafarevich, pp.11-13,26,65-66).
video   notes

Class 11 May 5, 2021: Field of fractions. Field extensions. Algebrically closed field. The fundamental theorem of algebra (without proof). Algebraic closure. Characteristic of a field. Finite fields. Prime subfield. GF(4). (Slides, pp.35-39; Lang, pp.89-90,99,107-108,110,244-245; Mac Lane-Birkhoff, pp.88,101-103,120-121,126; Shafarevich, pp.28-31; Vinberg, pp.20-22,129-130).
video   notes


7ALGS Algebraic Structures, summer semester 2019/2020

TOPICS, LITERATURE

moodle has a list of topics and pretty much good coverage of all of them, with mostly references to the literature in Czech. Here are some additional references to the English literature, compiled from my emails.

BONUS EXERCISES

Bonus exercises can be found in moodle, in the 4 pdf files entitled "BONUSOVÉ ÚLOHY".

Additionally, some bonus exercises were send by me in emails a while ago. Below, for your convenience, are reproduced those exercises which are not present in the files above.
  1. (up to 5 points for all groups) For any of the predefined groups in Group Explorer (all groups of order ≤ 20, and a few groups of higher order), specify whether it is: simple; solvable; nilpotent. If the group is solvable, determine its derived series (\(G^0 = G, G^n = [G^{n-1}, G^{n-1}]\)). If the group is nilpotent, determine its lower central series (\(G^0 = G, G^n = [G^{n-1}, G]\)).
  2. (0.5 points) Prove that any quotient of a solvable (nilpotent) group is solvable (nilpotent).
  3. (0.5 points each) P. Konečná, J. Kostra, M. Pomp, Grupy a okruhy, Ostravská univerzita, 2003, p. 80, Exercises 8, 9 (the book is available in moodle).

Created: Tue Feb 23 2021
Last modified: Mon Apr 29 18:40:41 CEST 2024