Syllabus
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Course Code: MMATH20- 101 Course Name: Abstract Algebra |
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| MODULE NO / UNIT | COURSE SYLLABUS CONTENTS OF MODULE | NOTES |
|---|---|---|
| 1 | Normal subgroup, quotient group, normalizer and centralizer of a non-empty subset of a group G, commutator subgroups of a group. first, second and third isomorphism theorems, correspondence theorem, Aut(G), Inn(G), automorphism group of a cyclic group, G-sets, orbit of an element in group G, Cayley’s theorem. conjugate elements and conjugacy classes, class equation of a finite group G and its applications, Burnside theorem. normal series, composition series, Jordan Holder theorem, Zassenhaus lemma, Scheier’s refinement theorem, solvable group, nilpotent group. (Chapter 5 and 6 of recommended book at Sr. No. 1, Chapter 5 of recommended book at Sr. No. 2) | |
| 2 | Cyclic decomposition, even and odd permutation, Alternation group An, simplicity of the Alternating group An (n>5). Cauchy’s theorem, Sylow’s first, second and third theorems and its applications to group of smaller orders. groups of order p2 and pq (q>p). (Chapter 7, 8.4 and 8.5 of recommended book at Sr. No 1) | |
| 3 | Modules, submodules, direct sums, finitely generated modules, cyclic module. R-homomorphism, quotient module, completely reducible modules, Schur’s lemma, free modules, representation of linear mapping, rank of linear mapping. (Chapter 14 of recommended book at Sr. No 1) | |
| 4 | Similar linear transformation, invariant subspaces of vector spaces, reduction of a linear transformation to triangular form, nilpotent transformation, index of nilpotency of a nilpotent transformation. Cyclic subspace with respect to a nilpotent transformations, uniqueness of the invariants of a nilpotent transformation. Primary decomposition theorem. Jordan blocks, Jordan canonical forms, cyclic module relative to a linear transformation, rational canonical form of a linear transformation and its elementary divisors, uniqueness of elementary divisors. (6.4. to 6.7 of recommended book of Sr. No. 3). | |
