Fall 2017 Math 602 Algebra - Preliminary Syllabus

I. Groups
1. Basics: groups and subgroups, normal subgroups, the subgroup lattice
2. Homomorphisms: quotient groups, homomorphism theorem, isomorphism theorems
3. Abelian groups: cyclic groups, direct products, structure theorem for finitely generated abelian groups
4. Permutation groups: group actions on sets, class equation, symmetric and alternating groups
5. Composition series: subnormal series and Jordan-Hoelder, solvable groups, nilpotent groups
6. Sylow theorems: Sylow subgroups, p-groups, groups of order pq, more on nilpotent groups
7. Semidirect products: definition and characterization, Schur-Zassenhaus theorem (relations to group cohomology), exact sequences


II. Rings
1. Rings and ideals: rings, ideals, prime and maximal ideals, homomorphisms, factor rings, principal ideal domains
2. Unique factorization domains
3. Extension and contraction of ideals
4. Nilradical and Jacobson radical


III. Linear Algebra
1. Vector Spaces: vector spaces, subspaces, intersection and sum, complements
2. Quotients: congruence relations, homomorphism theorem
3. Linear Operators: linear forms, dual spaces, duality theorem
4. Determinants: alternating forms, determinants
5. Eigenspaces: eigenvalues, minimal polynomial, primary decomposition, characteristic polynomial, Cayley-Hamilton
6. Jordan Canonical Form
7. Inner product spaces: orthogonality, hermitean forms, orthonomalization
8. Adjoints: self adjoint operators, isometries, normal operators, spectral theorem
9. Bilinear and quadratic forms
10. Multilinear algebra: tensor products, ring extensions, exterior powers, symmetric powers


Some of the material will be a review or more in-depth discussion of material you have seen in previous classes.