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 5:30-6:30pm, 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) 3.1-3.2 ZF Axioms 1-10 3.1.1, 3.1.3, 3.1.7, 3.1.10
7/1 -

Quiz 1

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8/1 (T) 3.3 Relations and functions 3.3.1, 3.4.2, 3.4.3, 3.4.4
10/1 (T) 3.4-3.6
8.1, 8.3		 Cardinalities of sets, countability 3.3.2, 3.3.5, 3.6.5, 3.6.6
8.1.2, 8.1.6, 8.3.3
3 13/1 (T) 8.5 Axiom of choice, Zorn's lemma 8.4.1, 8.5.3, 8.5.11, 8.5.18
14/1 -

Holiday

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15/1 (T) 4.1 - 4.4 Integers, rationals and gaps in them 4.1.3, 4.1.8, 4.2.6, 4.4.2
17/1 (B) 1.1-1.2
2.1-2.2		 Pigeonhole principles and mathematical induction Chap. 1: 2, 4, 9, 13
Chap. 2: 3, 5, 8, 11
4 20/1 (B) 3.1-3.3 Permutations, choices and the binomial theorem Chap. 2: 14, 28
Chap. 3: 1, 3, 6, 15, 20, 26
21/1 -

Quiz 2

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22/1 (B) 4.3, 5.1, 5.3 Combinatorial identities, compositions and partitions Chap. 4: 3, 5, 9, 18
Chap. 5: 1, 7, 11
24/1 (B) 5.2 Set partitions Chap. 5: 2, 4(a), 5, 16
5 27/1 (B) 6.1 Permutations by cycles Chap. 6: 2, 5, 8, 14, 17, 18, 22
28/1 -

Discussion

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29/1 (B) 7.1-7.2 Inclusion-Exclusion formulas Chap. 7: 1, 3, 5, 9, 13, 20
31/1 (B) 8.1 Ordinary generating functions Chap. 8: 1, 8, 9, 16
6 3/2

Class cancelled

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4/2 -

Quiz 3

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5/2 (B) 8.2 Exponential generating functions Chap. 8: 19, 20, 44, 45
7/2 (B) 9.1 Graph theory definitions and Eulerian tours Chap. 9: 2, 8, 12, 16
7 10/2 (B) 10.1 Trees and Cayley's formula Chap. 10: 2, 5, 7, 10, 15
11/2 -

Discussion

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12/2 (B) 10.4 Spanning trees and the matrix-tree theorem Chap. 10: Example 10.22, 16, 18(b, c), 20
14/2 (B) 12.1 Planar graphs Chap. 12: Example 12.4, Cor. 12.7, Prop. 12.8
8 17/2

Midsemester exam

 -
18/2 -

Exam week

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19/2

Exam week

 -
21/2

Exam week

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9 24/2 (IR) 1.1 Unique factorization in integers Chap. 1: 1, 6, 10, 11
25/2 -

Discussion

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26/2 (IR) 1.1-1.2 Unique factorization in polynomial rings Chap. 1: 13, 19, 20, 22
28/2 (IR) 1.3 Euclidean domains, PIDs and UFDs Chap. 1: 23, 25, 28, 31, 36, 37
10 3/3 (IR) 1.4, 2.1 Z[i] and Z[ω], Infinitude of primes and Dirichlet's theorem Chap. 1: 33, 34, 35
Chap. 2: 1, 4, 6, 16, 17
4/3 -

Quiz 4

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5/3 (IR) 2.2 Arithmetic functions Chap. 2: 9, 10, 13, 15, 22
7/3 (IR) 3.1-3.3 Congruences in Z Chap. 3: 1, 2, 3, 4, 8
11 10/3 (IR) 3.3-3.4 (Euler's, Fermat's little and Chinese remainder) theorems Chap. 3: 5, 12, 16, 23
11/3 -

Discussion

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12/3 (IR) 3.4, 4.1 Units in Zn Chap. 3: 17, 19, 20, 22
14/3 (IR) 4.1 Primitive roots Chap. 4: 2, 4, 5, 13, 14, 22
12 17/3 (IR) 5.1 Quadratic residues Chap. 5: 1, 2, 3, 14
18/3 -

Quiz 5

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19/3 (IR) 5.2-5.3
(DF) 1.1-1.2		 Law of quadratic reciprocity
Basic properties of groups		 Chap. 5: 1, 6, 7, 10
Sec. 1.1: 1, 9, 12, 13, 20
21/3 (DF) 1.3-1.6 Examples of groups, homomorphisms and isomorphisms Sec. 1.2: 4, 7, 9, 10, 18
Sec. 1.3: 4, 10, 11, 15
Sec. 1.6: 1, 2, 5, 11, 18
13 24/3 (DF) 1.7, 2.1 Group actions
Subgroups		 Sec. 1.7: 2, 6, 8, 11, 15
Sec. 2.1: 1, 3, 5, 9
25/3 -

Discussion

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26/3 (DF) 2.2-2.3 Centralizers, normalizers and stabilizers Sec. 2.2: 2, 3, 5(a), 6(a), 10
28/3 (DF) 2.3, 3.1 Cyclic groups and their subgroups Sec 2.3: 2, 12, 16, 20, 24
14 31/3

Holiday

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1/4 -

Quiz 6

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2/4 (DF) 3.1 Quotient groups and cosets Sec 3.1: 2, 6, 9, 11, 12
4/4 (DF) 3.1, 3.2 Normal subgroups and Lagrange's theorem Sec 3.1: 14(a, b), 22(a), 26(a, b), 36
Sec 3.2: 4, 5, 6, 10, 16, 22
15 7/4 (DF) 3.2-3.3 First isomorphism theorem Sec 3.2: 1, 2, 11
Sec 3.3: 1, 8
8/4 -

Quiz 7

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9/4 (DF) 4.1-4.2 Group actions and Cayley's theorem Sec 4.1: 1, 4, 6, 9(a)
Sec 4.2: 1, 2, 9, 11
11/4 (DF) 4.3 Conjugation action and class equation Sec 4.3: 2, 4, 10, 12, 13
19 23/4 -

Final Exam