Advance Group Theory-II (MATH-616 & MATH-408)

Discription

This course is the continuation of the course "Advanced Group Theory-I" and covers the advanced topics in group theory such as solvable groups, nilpotent groups, and linear groups.

Pre-Requisite

Advance Group Theory-I

Learning Outcomes

On successful completion of this course students will be able to:

define solvable groups, nilpotent groups
derive upper and lower central series, derived series and representation of linear groups
reason mathematically, to write simple proofs, and can judge when an attempted proof in group theory is correct/complete or is not.
have a chance to reflect on doing mathematics, solving problems and on our role and progress as mathematicians.

Series in groups, normal series, normal series and its refinement
Composition series, equivalent composition series
Jordan Holder Theorem
Solvable groups, definition, examples and related results
Upper and Lower Central series, Nilpotent groups
Characterization of finite nilpotent groups
The Frattini subgroups, definition, examples and related results
Free groups, definition, examples and related results
Free Product, definition, examples and related results
Group algebras and representation modules

Recommended Books

Rotman J. J., An Introduction to the theory of groups, (Springer; 4th edition, 1999).
Shah S. K., Shankar A. G., Group Theory, (Dorling Kindersley 2013).

Suggested Books

Bhattacharya P. B, Jain S. K. and Nagpaul S. R. Basic Abstract Algebra (Cambridge University Press, 2nd edition, 1994)
Fraleigh J. B. A First Course in Abstract Algebra, (Addison-Wesley Publishing Company, 7th edition, 2003).
Malik D. S., Mordeson J. N. and Sen M. K. Fundamentals of Abstract Algebra, (WCB/McGraw-Hill, 1997).
Rose J., A Course on Group Theory, (Dover Publications; Revised edition 2012).