UM 205: Introduction to Algebraic Structures

Instructor: Arvind Ayyer
Office: X-15 (new wing)
Office hours: Tuesday, 4:30-5:30pm
Phone number: (2293) 3215
Email: (First name) at iisc dot ac dot in
Class Timings: Mondays, Wednesdays and Fridays, 11:00 — 11:50am
Classroom: G - 21, OPB
To join the course on MS Teams, use the code aicr78k
Textbooks:
  • T. Tao, Analysis I, 3rd edition, Hindustan Book Agency.
  • L. Childs, A Concrete Introduction to Higher Algebra, 3rd edition, Springer-Verlag.
  • M. A. Armstrong, Groups and Symmetry, Springer-Verlag.
  • Miklos Bona, A Walk Through Combinatorics: An Introduction to Enumeration and Graph Theory, 3rd edition, World Scientific.
  • G. Fraleigh, A First Course in Abstract Algebra, 7th edition, Pearson.
  • K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Springer-Verlag
  • D. S. Dummit and R. M. Foote, Abstract Algebra, 3rd Edition, Wiley
TAs and office hours:

Email ids are in paranthesis and end with at iisc dot ac dot in, and offices are in the maths department.

  • Ritabrata Das (ritabratadas), M 4-5pm, L26
  • Ashutosh Jangle (ashutoshjs), W 4-5pm, X20
  • Lalit Thuwal (lalitthuwal), S 3-4pm, N04
  • Manpreet Singh (manpreets), Th 4-5pm, N11
  • Kamla Kant Mishra (kamlak), F 4-5pm, L12
Tutorials: Tuesdays 9:00 — 9:50am, G-01 and G-21

Course Description

  1. Set theory: Peano axioms, ZFC axioms, axiom of choice/Zorn's lemma, countable and uncountable sets.
  2. Combinatorics: induction, pigeonhole principle, inclusion-exclusion, posets and the Möbius inversion formula, recurrence relations.
  3. Graph theory: Basics, trees, Eulerian tours, matchings, matrices associated to graphs.
  4. Number theory: Unique factorization in Z and k[x], rings, fields, unique factorization in a PID, the ring Z[i], Infinitude of primes, primes in arithmetic progression, Arithmetic functions, Congruences, Euler's theorem, Fermat's little theorem, Chinese remainder theorem, Structure of units in Z/NZ, quadratic reciprocity.
  5. Algebra: Definition of groups, Examples (dihedral groups, symmetric groups, cyclic groups), subgroups, homomorphisms and isomorphisms, cyclic groups and its subgroups, quotient groups, cosets, Lagrange's theorem, first isomorphism theorem, group actions, Cayley's theorem, class equation.

Exams

All exams will be closed book, closed notes, and
no calculators or electronic devices are allowed (no cell/smart phones).
No communication among the students will be tolerated.
There will be no make up exams.

The date for the midterms and final will be announced later.

Grading

Here are the weights for the homework and exams.
All marks will be posted online on Teams.

Tentative Class Plan

Tutorials are marked in green.

week date sections material covered homework and other notes
1 1/1 (T) 2.1 Peano axioms No homework :)
3/1 (T) 2.2-2.3 Basic axioms 2.2.1, 2.2.4, 2.2.6, 2.3.2, 2.3.4
2 6/1 (T) 3.1-3.3, 3.4 Relations and functions
7/1 -

Quiz 1

 -
8/1 (T) 3.3, 3.6,
8.1, 8.3		 Cardinalities of sets, countability
10/1 (T) 8.4, 8.5 Axiom of choice, Zorn's lemma
3 13/1 (T) 4.1 - 4.4 Integers, rationals and gaps in them
14/1 -

Holiday

 -
15/1 (B) 1.1-1.2
2.1-2.2		 Pigeonhole principles and mathematical induction
17/1 (B) 3.1-3.3 Permutations, choices and the binomial theorem
4 20/1 (B) 4.3, 5.1, 5.3 Combinatorial identities, compositions and partitions
21/1 -

Quiz 2

 -
22/1 (B) 5.2, 6.1 Set partitions
24/1 (B) 7.1-7.2 Permutations by cycles
5 27/1 (B) 8.1 Inclusion-Exclusion formulas and
Ordinary generating functions
28/1 -

Discussion

 -
29/1 (B) 8.2 Exponential generating functions
31/1 (B) 9.1 Graph theory definitions and Eulerian tours
6 3/2

Class cancelled

 -
4/2 -

Quiz 3

 -
5/2 (B) 10.1 Trees and Cayley's formula
7/2 (B) 10.4 Spanning trees and the matrix-tree theorem
7 10/2 (B) 12.1 Planar graphs
11/2 -

Discussion

 -
12/2 (IR) 1.1 Unique factorization in integers
14/2 -

Exam week

 -
8 17/2

Midsemester exam

 -
18/2 -

Exam week

 -
19/2

Exam week

 -
21/2

Exam week

 -
9 24/2 (IR) 1.1-1.2 Unique factorization in polynomial rings
25/2 -

Discussion

 -
26/2 (IR) 1.3-1.4 Principal ideal domains, Z[i] and Z[ω]
28/2

Class cancelled

 -
10 3/3 (IR) 2.1 Infinitude of primes and Dirichlet's theorem
4/3 -

Quiz 4

 -
5/3 (IR) 2.2 Arithmetic functions
7/3 (IR) 3.1-3.3 Congruences in Z
11 10/3 (IR) 3.3-3.4 Euler's theorem, Fermat's little theorem and Chinese Remainder Theorem
11/3 -

Discussion

 -
12/3 (IR) 4.1 Units in Zn
14/3 (IR) 5.1 Quadratic residues
12 17/3 (IR) 5.2-5.3 Law of quadratic reciprocity
18/3 -

Quiz 5

 -
19/3 (DF) 1.1-1.2 Basic properties of groups
21/3 (DF) 1.3-1.6 Examples of groups, homomorphisms and isomorphisms
13 24/3 (DF) 1.7, 2.1 Group actions
Subgroups
25/3 -

Discussion

 -
26/3 (DF) 1.7, 2.1 Centralizers, normalizers and cyclic groups
28/3 (DF) 2.3, 31 Subgroups of cyclic groups and quotient groups
14 31/3 (DF) 2.2-2.3

Holiday

 -
1/4 -

Quiz 6

 -
2/4 (DF) 3.1 Quotient groups and cosets
4/4 (DF) 3.2, 3.3 Lagrange's theorem and first isomorphism theorem
15 7/4 (DF) 4.1-4.3 Cayley's theorem and class equation
8/4 -

Discussion

 -
9/4
11/4
19 ?/4 -

Final Exam