UM 205: Introduction to Algebraic Structures

Instructor: Arvind Ayyer
Office: X-15 (new wing)
Office hours: Tuesday, 4-5pm
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 gmw03ii
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
  • Srijan Sarkar (srijans), W 4-5pm, X13
  • Manpreet Singh (manpreets), Th 5-6pm, N11
  • Srikanth Pai (srikanthpai), F 2-3pm, N11
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

Class cancelled

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2/1 -

Class cancelled

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3/1 (T) 2.1-2.3 Peano axioms No homework :)
5/1

Class cancelled

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2 8/1 (T) 3.1 Basic axioms 2.2.1, 2.2.4, 2.2.6, 2.3.2, 2.3.4
3.1.1, 3.1.3, 3.1.7, 3.1.10
9/1 -

Discussion

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10/1 (T) 3.3, 3.4 Relations and functions 3.4.1, 3.4.3, 3.4.4
12/1 (T) 3.3, 3.6,
8.1, 8.3		 Cardinalities of sets, countability 3.3.2, 3.3.5, 3.6.7, 3.6.6
8.1.2, 8.1.6, 8.3.3
3 15/1 (T) 8.4, 8.5 Axiom of choice, Zorn's lemma 8.4.1, 8.5.3, 8.5.11, 8.5.18
16/1 -

Quiz 1

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17/1 (T) 4.1 - 4.4 Integers, rationals and gaps in them 4.1.3, 4.1.8, 4.2.6, 4.4.2(a)
19/1 (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 22/1 (B) 3.1-3.3 Permutations, choices and the binomial theorem Chap. 2: 14, 28
Chap. 3: 1, 3, 7, 15, 21
23/1 -

Discussion

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24/1 (B) 4.3, 5.1, 5.3 Combinatorial identities, compositions and partitions Chap. 4: 3, 5, 9, 18
Chap. 5: 1, 7, 11
26/1

Holiday

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5 29/1 (B) 5.2, 6.1 Set partitions Chap. 5: 2, 4(a), 5, 16
30/1 -

Quiz 2

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31/1 (B) 7.1-7.2 Permutations by cycles Chap. 6: 2, 5, 8, 14, 17, 18, 22
2/2 (B) 8.1 Inclusion-Exclusion formulas and
Ordinary generating functions		 Chap. 7: 1, 9, 13
Chap. 8: 1, 2, 8, 9
6 5/2 (B) 8.2 Exponential generating functions Chap. 8: 3, 16, 19, 20
6/2 -

Discussion

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7/2 (B) 9.1 Graph theory definitions and Eulerian tours Chap. 9: 2, 8, 12, 16
9/2 (B) 10.1 Trees and Cayley's formula Chap. 10: 2, 5, 7, 10, 15
7 12/2 (B) 10.4 Spanning trees and the matrix-tree theorem Chap. 10: Example 10.22, 16, 18(b, c), 20
13/2 -

Quiz 3

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14/2 (B) 12.1 Planar graphs Chap. 11: Example 12.4, Cor. 12.7, Prop. 12.8
16/2 -

Exam week

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8 19/2

Midsemester exam

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20/2 -

Exam week

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21/2

Exam week

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23/2

Exam week

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9 26/2 (IR) 1.1 Unique factorization in integers Chap. 1: 1, 6, 10, 11
27/2 -

Discussion

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28/2 (IR) 1.1-1.2 Unique factorization in polynomial rings Chap. 1: 7, 13, 19, 20
1/3 (IR) 1.3-1.4 Principal ideal domains, Z[i] and Z[ω] Chap. 1: 31, 34, 36, 37
10 4/3 (IR) 2.1 Infinitude of primes and Dirichlet's theorem Chap. 1: 33, 35
Chap. 2: 1, 4, 6, 16, 17
5/3 -

Quiz 4

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6/3 (IR) 2.2 Arithmetic functions Chap. 2: 9, 10, 11, 13, 15, 22
8/3

Holiday

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11 11/3 (IR) 3.1-3.3 Congruences in Z Chap. 3: 2, 3, 4, 8
12/3 -

Discussion

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13/3 (IR) 3.3-3.4 Euler's theorem, Fermat's little theorem and Chinese Remainder Theorem Chap. 3: 5, 12, 16, 23
15/3 (IR) 4.1 Units in Zn Chap. 4: 2, 4, 5, 13, 14, 22
12 18/3 (IR) 5.1 Quadratic residues Chap. 5: 2, 3, 6, 14, 15
19/3 -

Quiz 5

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20/3 (IR) 5.2-5.3 Law of quadratic reciprocity Chap. 5: 1, 6, 7, 10, 12, 16(i)
22/3 (DF) 1.1-1.2 Basic properties of groups Sec. 1.1: 1, 5, 9, 12, 13, 20
Sec. 1.2: 4, 9, 10
13 25/3 (DF) 1.3-1.6 Examples of groups, homomorphisms and isomorphisms Sec. 1.2: 7, 18
Sec. 1.3: 4, 10, 11, 15
Sec. 1.6: 1, 2, 5, 11, 18
26/3 -

Discussion

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27/3 (DF) 1.7, 2.1 Group actions
Subgroups		 Sec. 1.7: 2, 6, 8, 11, 15
Sec. 2.1: 1, 3, 5, 9
29/3

Holiday

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14 1/4 (DF) 2.2-2.3 Centralizers, normalizers and cyclic groups Sec. 2.2: 2, 3, 5(a), 6(a), 10
Sec. 2.3: 2, 12, 16
2/4 -

Quiz 6

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3/4 (DF) 2.3, 31 Subgroups of cyclic groups and quotient groups Sec 2.3: 20, 24
Sec 3.1: 1, 2, 6, 9, 11
5/4 (DF) 3.1 Quotient groups and cosets Sec 3.1: 12, 14(a, b), 22(a), 24, 26(a, b), 36
15 8/4 (DF) 3.2, 3.3 Lagrange's theorem and first isomorphism theorem Sec 3.2: 4, 5, 6, 10, 15
9/4 -

Discussion

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10/4 (DF) 4.1-4.3 Cayley's theorem and class equation Sec 4.1: 1, 4, 9(a)
Sec 4.2: 1, 2, 9, 11
Sec 4.3: 2, 4, 10
19 22/4 -

Final Exam