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.