Schedule

[A] Ash,

[H] Hungerford,

[OS] O'Sullivan,

[JB] Judson, Beezer

Free groups.

Generators and relations

Day | Topics | Preparation |
---|---|---|

Th. 1/19 | Introduction: Groups, Rings, Fields. Groups, subgroups and homomorphisms. |
[OS] Sec. 1. |

Tu. 1/26 | Order theorem. Automorphism group. Lattice of subgroups, Symmetric Group. |
[OS] Sec. 1-2 |

Th. 1/26 | Groups in sage. Meet in lab GMCS 425 |
[JB] Ch 3-6 Sage exercises. |

Tu. 1/31 | Alternating group Cosets, Index of a group,Cayley's theorem. Conjugation. |
[OS] Sec. 2-3 |

Th. 2/2 | Rings and unit groups. | [OS]. |

Tu. 2/7 | Irreducibility in polynomial rings.
Quotients of . F[x] |
[OS]. |

Th. 2/9 | Cubic field extensions. | [OS]. |

Tu. 2/14 | Finite Fields. | [OS]. |

Th. 2/16 | Automorphisms of Finite Fields: The Frobenius map. | [OS]. |

Tu. 2/21 | Irreducible over the rationals, Q Quadratic extensions of the rationals. |
[OS] |

Th 2/23 | Roots of a cubic equation
and field extensions. |
[OS] |

Tu 2/28 | Normal subgroups and quotient groups The factor theorem. |
[A] Sec. 1.2-3 |

Th 3/1 | The 1st and 3rd isomorphism theorems and the correspondence
theorem Matrix groups examples. |
[A] Sec. 1.4 |

Tu 3/6 | The second isomorphism theorem. Direct and semidirect products. | [A] Sec. 1.4-5 |

Th 3/8 | Direct and semidirect products. | [A] Sec. 1.4-5 [A] Sec. 5.8 |

Tu. 3/13 | Groups acting on a set. | [A] 5.1-2. |

Th. 3/15 | Orbit/stabilizer theorem. Groups of prime-power order First Sylow theorem. |
[A] 5.1-2. |

Tu.3/20 | The second and third Sylow theorem. Application of Sylow theorems. |
[A] 5.4-5. |

Th. 3/22 | Abelian groups. | [H] 8.2 (or other sources). |

Tu. 4/3 | Algebraic field extensions | [A] 3.1-2; [H] 10.4. |

Th. 4/5 | Some group theory problems. Algebraic extensions and splitting fields. |
[A] 3.1-2; [H] 10.4. |

Tu. 4/10 | TEST: Groups. | . |

Th. 4/12 | Splitting fields and algebraic closure. Separable extensions. |
[A] 3.3-5 |

Tu 4/17 | Primitive element Theorem. Normal and Galois extensions. The Galois correspondence. |
[A] 6.1-2; [H] 12.2. |

Th 4/19 | The fundamental theorem of Galois theory. | [A] 6.2; [H] 12.2. |

Th 4/24 | Cyclotomic fields extensions. | [A] 6.3,6. [A] 6.5 |

Th 4/26 | Cyclic and abelian extensions. | [A] 6.7. |

Tu 5/1 | Geoemtric constructions. | [A] 6.8. |

Th. 5/3 | Commutator subgroups and solvable groups. | [A] 5.7. |

Tu. 5/8 | Solvability by radicals | [A] 6.8 |