BSc (HONS) V semester Group theory -II :

C12 Group Theory-II

Total Marks: 100

Theory: 75

Internal Assessment: 25

5 Lectures, 1 Tutorial (per week per student)

Automorphism, inner automorphism, automorphism groups, automorphism groups of finite and infinite cyclic groups, applications of factor groups to automorphism groups, Characteristic subgroups, Commutator subgroup and its properties.

[1]: Chapter 6, Chapter 9 (Theorem 9.4), Exercises 1-4 on page168, Exercises 52, 58 on page Pg 188.

Properties of external direct products, the group of units modulo n as an external direct product, internal direct products, Fundamental Theorem of finite abelian groups.

[1]: Chapter 8, Chapter 9 (Section on internal direct products), Chapter 11.

Group actions, stabilizers and kernels, permutation representation associated with a given group action, Applications of group actions: Generalized Cayley’s theorem, Index theorem. Groups acting on themselves by conjugation, class equation and consequences, conjugacy in Sn, p-groups, Sylow’s theorems and consequences, Cauchy’s theorem, Simplicity of An for n ≥ 5, non-simplicity tests.

[2]: Chapter 1 (Section 1.7), Chapter 2 (Section 2.2), Chapter 4 (Section 4.1-4.3, 4.5- 4.6).

[1]: Chapter 25.

REFERENCES:

1. Joseph A. Gallian, Contemporary Abstract Algebra (4th Ed.), Narosa Publishing House, 1999.

2. David S. Dummit and Richard M. Foote, Abstract Algebra (3rd Edition), John Wiley and Sons (Asia) Pvt. Ltd, Singapore, 2004

NOTE: In coming summer I will be starting the notes for this course. I will be away from 15th July 2019 - 3rd August 2019 ( attending a research-workshop in ICTS, Bengaluru). My classes will start from 4th August 2019.

INSTRUCTION: From 19th July 2019 - 3rd August 2019 , I request you all to revise your 3rd Semester GROUP THEORY-I. In Group Theory-II we are going to use many results of Group Theory - I.

VERY IMPORTANT: On 4th August 2019 everyone should have their personal copies of following books (A must):

• Joseph A. Gallian, Contemporary Abstract Algebra (4th Ed.), Narosa Publishing House, 1999
• David S. Dummit and Richard M. Foote, Abstract Algebra (3rd Edition), John Wiley and Sons (Asia) Pvt. Ltd, Singapore, 2004

* Those who can not afford the books will get it issued from Departmental Library whose incharge is Dr Dinesh Kumar ( you have to give valid reasons for that).

** No one will be allowed in the class without these books. XEROX and PHOTOCOPIES NOT ALLOWED.

*** Tutorial classes are the most important part of this course. Here you can ask questions, clarify the concepts you missed in the regular class, assignments will be discussed because here students-group size is small.

**** I will post the details of Internal assessment before the start of the class.

*****Those of you thinking about research as career , here is a Journal , some portion of it may be accessible to you : http://amj.math.stonybrook.edu/