THIRD SEMESTER MATHEMATICS
ABSTRACT ALGEBRA
Course Outcomes:
After successful completion of this course, the student will be able to;
1.acquire the basic knowledge and structure of groups, subgroups and cyclic groups.
2. get the significance of the notation of a normal subgroups.
3. get the behavior of permutations and operations on them.
4. study the homomorphisms and isomorphisms with applications.
5.understand the ring theory concepts with the help of knowledge in group theory and to prove the theorems.
6. understand the applications of ring theory in various fields.
Course Syllabus:
UNIT – I
GROUPS :Binary Operation – Algebraic structure – semi group-monoid – Group definition and elementary properties Finite and Infinite groups – examples – order of a group, order of an element, Composition tables with examples.
UNIT – II
SUBGROUPS :Complex Definition – Multiplication of two complexes Inverse of a complex-Subgroup definition- examples-criterion for a complex to be a subgroups. Criterion for the product of two subgroups to be a subgroup-union and Intersection of subgroups.Co-sets and Lagrange’s Theorem :Cosets Definition – properties of Cosets–Index of a subgroups of a finite groups–Lagrange’s Theorem.
UNIT –III
NORMAL SUBGROUPS :Definition of normal subgroup – proper and improper normal subgroup–Hamilton group – criterion for a subgroup to be a normal subgroup – intersection of two normal subgroups – Sub group of index 2 is a normal sub group –quotient group – criteria for the existence of a quotient group.
HOMOMORPHISM :Definition of homomorphism – Image of homomorphism elementary properties ofhomomorphism – Isomorphism – automorphism definitions and elementary properties–kernel of a homomorphism – fundamental theorem on Homomorphism and applications.
UNIT – IV
PERMUTATIONS AND CYCLIC GROUPS :Definition of permutation – permutation multiplication – Inverse of a permutation – cyclic permutations – transposition – even and odd permutations – Cayley’s theorem.Cyclic Groups :- Definition of cyclic group – elementary properties – classification of cyclic groups.
UNIT – V
RINGS :Definition of Ring and basic properties, Boolean Rings, divisors of zero and cancellation laws Rings, Integral Domains, Division Ring and Fields, The characteristic of a ring - The characteristic of an Integral Domain, The characteristic of a Field. Sub Rings, Ideals