Course Syllabus
This syllabus outlines the modules of Advanced Algebra.
- Group and their properties; Permutation groups
- Group actions; orbit-stabilizer theorem; class equation; conjugacy action
- group action on cosets; permutation representations
- applications to finite groups
- Sylow theorems with applications
- simplicity and non-simplicity tests
- Normal series; subnormal series; composition series; Jordan-Holder theorem; derived series
- solvable groups; nilpotent groups
- lower and upper central series; finite p-groups
- relation between nilpotent and solvable groups; simple groups
- Prime ideals and maximal ideals; nilpotent elements and nilradical; Jacobson radical
- Euclidean domains; principal ideal domains; unique factorization domains
- irreducible and prime elements; relation among ED, PID and UFD
- Noetherian rings; Field and extension field
- Modules; submodules; quotient modules
- module homomorphisms; kernel and image; exact sequences
- direct sums and direct products; free modules; cyclic modules; finitely generated modules
- simple and semisimple modules: introductory examples
- Torsion elements and torsion submodules; annihilators
- cyclic decomposition
- structure theorem for finitely generated modules over a PID
- invariant factor form; elementary divisor form
- applications to finitely generated abelian groups