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GROUPS AND RINGS (Code-09050502)

S.No. Topic Domain Hours as per UGC
1 ·         Definition of a group with example and simple properties of groups Must know 10
·         Subgroups and subgroup criteria
·         Generation of groups, cyclic groups
·         Cosets, Left and right cosets
·         Index of a sub-group, Coset decomposition
·         Lagrange’s theorem and its consequences
·         Normal subgroups, Quotient groups
2 ·         Homomorphisms, isomorphisms Must know 15
·         Automorphisms and inner automorphisms of a group
·         Automorphisms of cyclic groups
·         Permutations groups, even and odd permutations
·         Alternating groups
·         Cayley’s theorem
·         Centre of a group and derived group of a group
3 ·         Introduction to rings, subrings Nice to know 15
·         Integral domains and fields
·         Characteristics of a ring, Ring homomorphism
·         Ideals(principle, prime and Maximal) and Quotient rings
·         Field of quotients of an integral domain
4 ·         Euclidean rings Must know 12
·         Polynomial rings
·         Polynomials over the rational field
·         The Einstein’s criterion
·         Polynomial rings over commutative rings
·         Unique factorization domain
·         R unique factorization domain implies so is R[X1,X2,…XN]

Books Recommended:

  1. N Herstein : Topics in Algebra, Wiley Eastern Ltd., New Delhi, 1975
  2. B. Bhattacharya, S.K. Jain and S.R. Nagpal : Basic Abstract Algebra (2nd edition).
  3. Vivek Sahai and Vikas Bist : Algebra, NKarosa Publishing House.
  4. S.Luther and I.B.S. Passi : Algebra, Vol. II, Narosa Publishing House.
  5. John B. Fraleigh, A First course in Abstract Algebra, 7th, Pearson, 2002.
  6. Artin, Abstract Algebra, 2nd Ed., Pearson, 2011.
  7. Joseph A. Gallian, Contemporary Abstract Algebra, 4th, Narosa Publishing House, 1999.
  8. David S. Dummit and Richard M. Foote, Abstract Algebra, 3rd, John Wiley and sons (Asia) Pvt. Ltd., Singapore, 2004.
  9. R. Durbin, Modern Algebra, John Wiley and sons, New York inc., 2000.
  10. D. A. R. Wallace, Groups, Rings and Fields, Springer Verlag London Ltd., 1998.