Utkal University, Odisha, Mathematics Semester 6, GROUP THEORY II Syllabus

GROUP THEORY II

UNIT-I
Automorphism, inner automorphism, automorphism groups, automorphism groups of finite and infinite cyclic groups, applications of factor groups to automorphism groups. characteristic
subgroups.

UNIT-II
Commutator subgroup and its properties, 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.

UNIT-III
Group actions, stabilizers and kernels, permutation representation associated with a given group action, Application of group actions: Generalized Cayley's theorem, Index theorem.

UNIT-IV
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.

BOOKS RECOMMENDED :
1. John B. Fraleigh, A First Course in Abstract Algebra, Narosa Publishing House, New Delhi.
2. Joseph A. Gallian Contemporary Abstract Algebra (4th Edition), Narosa Publishing House, New Delhi.

BOOK FOR REFERENCES:
1. M. Artin, Abstract Algebra, 2nd Ed., Pearson, 2011.
2. David S. Dummit and Richard M. Foote, Abstract Algebra, 3rd Ed., John Wiley and Sons (Asia) Pvt. Ltd., Singapore, 2004.
3. J.R. Durbin, Modern Algebra, John Wiley & Sons, New York Inc., 2000.