Mathematics

**MTH ****301: Groups Theory**** ****(4)**

*Learning Objectives*:

This is an introductory course on Group theory. We will begin by studying the basic concepts of subgroups, homomorphisms and quotient groups with many examples. We then study group actions, and prove the Class equation and the Sylow theorems. They are in turn used to prove the structure theorem for finite abelian groups and to discuss the classification of groups of small order. We then turn to solvability, prove the Jordan-Holder theorem, and discuss nilpotent groups (if time permits).

*Course Contents*:

- Definition of group, basic properties, examples (Dihedral, Symmetric, Groups of Matrices, Quaternion Group, Cyclic, Abelian Groups)
- Homomorphisms, Isomorphisms, subgroups, subgroup generated by a set, subgroups of cyclic groups
- Review of Equivalence relations, Cosets, Lagrange’s theorem, Normal subgroup, Quotient Group, Examples, Isomorphism theorems, Automorphisms
- Group actions, orbits, stabilizer, faithful and transitive actions, centralizer, normalizer, Cayley’s theorem, Action of the group on cosets
- Conjugation, Class equation, Cauchy’s theorem, Applications to p-groups, Conjugacy in
_{}S_{n} - Sylow theorems, Simplicity of A
_{n}and other applications - Direct products, Structure of Finite abelian groups
- Semi-Direct products, Classification of groups of small order
- Normal series, Composition series, Solvable groups, Jordan-H¨older theorem, Insolvability of S
_{5} - Lower and upper central series, Nilpotent groups, Basic commutator identities, Decomposition theorem of finite nilpotent groups (if time permits)

*Suggested Books*:

- I. N. Herstein, Topics in Algebra, 2
^{nd}Edition, Wiley, 2006 - T. W. Hungerford, Algebra, Springer Verlag, 2005
- M. Artin, Algebra, Prentice-Hall of India, 1994
- D. S. Dummit, R. M. Foote, Abstract Algebra, 2
^{nd}Edition, Wiley - J. Rotman, A First Course in Abstract Algebra : With Applications, Prentice Hall
- J. Rotman, An Introduction to Theory of Groups, Springer GTM, 1999
- H. Kurzweil, B. Stellmacher, The Theory of Finite Groups, Springer Universitext, 2004
- M. Suzuki, Group Theory I, Springer GMW 247

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