SEMESTER-III, PAPER-III
CBCS/ SEMESTER SYSTEM (w.e.f. 2020-21 Admitted Batch)
B.A./B.Sc. MATHEMATICS
ABSTRACT ALGEBRA SYLLABUS (75 Hours)
Course Outcomes:
After successful completion of this course, the student will be able to;
- acquire the basic knowledge and structure of groups, subgroups and cyclic
- get the significance of the notation of a normal
- get the behavior of permutations and operations on
- study the homomorphisms and isomorphisms with
- understand the ring theory concepts with the help of knowledge in group theory and to prove the theorems.
- understand the applications of ring theory in various
Course Syllabus:
UNIT – I (12 Hours)
GROUPS :
Binary Operation – Algebraic structure – semi group-monoid – Group definition and elementary properties Finite and Infinite groups – examples – order of a group, Composition tables with examples.
UNIT – II (12 Hours)
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(Statements only)–Index of a subgroups of a finite groups–Lagrange’s Theorem.
UNIT –III (12 Hours)
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.
HOMOMORPHISM:
Definition of homomorphism – Image of homomorphism elementary properties of homomorphism – Isomorphism – automorphism definitions –kernel of a homomorphism – fundamental theorem on Homomorphism.
UNIT – IV (12 Hours)
PERMUTATION GROUPS:
Definition of permutation – permutation multiplication – Inverse of a permutation – cyclic permutations – transposition – even and odd permutations – Cayley’s theorem.
UNIT – V (12 Hours)
RINGS:
Definition of Ring and basic properties, Boolean Rings, divisors of zero and cancellation laws Rings, Integral Domains, Division Ring and Fields, Definitions and examples only in Sub Rings, Ideals.
Co-Curricular Activities(15 Hours)
Seminar/ Quiz/ Assignments/ Group theory and its applications / Problem Solving.
Text Book :
A text book of Mathematics for B.A. / B.Sc. by B.V.S.S. SARMA and others, published by S.Chand & Company, New Delhi.
Reference Books :
- Abstract Algebra by J.B. Fraleigh, Published by Narosa publishing
- Modern Algebra by M.L.
Rings and Linear Algebra by Pundir & Pundir, published by Pragathi Prakashan
