| 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 |
| 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 |