| Day | Topics | Preparation |
|---|---|---|
| Wed. 1/20 | Introduction. Groups: definition, dihedral and permutation (symmetric) groups. |
Sec. 7.1-2 |
| Fri. 1/22 | First properties of groups.
Order of an element. |
Sec. 7.1-2 |
| Mon. 1/25 | Subgroups: treated abstractly. | Sec. 7.3 |
| Wed. 1/27 | Groups from rings. (Z_n, +) , U_n , Gl(2,n) , and F* , for F a field. |
Sec. 7.2-3 |
| Fri. 1/29 | Matrix groups and subroups. | Sec. 7.2-3 |
| Mon. 2/1 | Homomorphisms and related subgroups. | Sec. 7.3-4 |
| Wed. 2/3 | Isomorphisms and automorphisms. | Sec. 7.4 |
| Fri. 2/5 | Group theory in SAGE. | Sec. 7.1-4 |
| Mon. 2/8 | Cayley's theorem. | Sec. 7.4,5 |
| Wed. 2/10 | Lagrange's theorem. Cosets and congruence. |
Sec. 7.5 | Fri. 2/12 | Matrix groups, cosets, etc., in Sage. | Sec. 7.1-6 |
| Mon. 2/15 | Classification of groups of order <8. Normal subgroups. |
Sec. 7.6 |
| Wed. 2/17 | Normal subgroups theorems. | Sec. 7.7 |
| Fri. 2/19 | Survey of exercises. Quotient groups. | Sec. 7.6-8 |
| Mon. 2/22 | The three isomorphism theorems. | Sec. 7.6-8 |
| Wed. 2/24 | More on the isomorphism theorems. | Sec. 7.6-8 |
| Fri. 2/26 | Miscellaneous problems The groups D_n . |
Sec. 8.1 |
| Mon. 3/1 | D_n and groups of order 2p | Sec. 7.1-8 and p. 277-279 |
| Wed. 3/3 | EXAM. | Sec. 7.1-8 |
| Fri. 3/5 | Direct products | Sec. 8.1 |
| Mon. 3/8 | Internal direct product theorem. Finite abelian groups | Sec. 7.7,8 Sec. 8.1 |
| Wed. 3/10 | Fundamental Theorem of finite abelian groups. | Sec. 8.2 |
| Fri. 3/12 | Some exercises Q/Z, R/Z and C* . | Sec. 7.7-8 |
| Mon 3/15 | Factorization of abelian p -groups. | Sec. 8.2 |
| Wed. 3/17 | Invariant factor decomposition. | Sec. 8.2 |
| Fri. 3/19 | The symmetric group, S_n . | Sec. 7.9 |
| Mon. 3/22 | The symmetric group, S_n and the alternating group A_n . |
Sec. 7.9 |
| Wed. 3/24 | The alternating group, A_n, is simple. | Sec. 7.10 |
| Fri. 3/26 | Furlough day. | |
| Mon. 4/5 | Statement of the Sylow theorems. Application to dihedral groups. |
Sec. 8.3-5 |
| Wed. 4/7 | Permutation groups and M.O'S graduating class. The probability of no large cycle. | . |
| Fri. 4/9 | EXAM. | Sec. 7.6-10, 8.1-4 |
| Mon. 4/12 | Conjugacy and a proof of the first Sylow theorem. | Sec. 8.4 |
| Wed. 4/14 | Simple groups and the Sylow theorems. | Sec. 8.5, Sec 8.3 #1-7 |
| Fri. 4/16 | Sage exercises. | . |
| Mon. 4/19 | Polynomial rings over a field.
Division and the GCD. Roots and remainders. Irreducibles and unique factorization. |
Ch 4 |
| Wed. 4/21 | Polynomial rings modulo a polynomial. | Ch. 5. |
| Fri. 4/23 | No class: work on your projects. | . |
| Mon. 4/26 | Roots and factors, irreduciblity. Adjoining roots.. |
Sec. 4.4,5, 5.3 |
| Wed. 4/28 | Extensions of the rationals. Finite fields and the Freshman's rule. |
Ch. 5 |
| Fri. 4/30 | We meet to discuss projects. | . |
| Mon. 5/3 | The main theorem of finite fields. F_16 = F_2[x]/ x^4+ x +1 in all its glory. |
Ch. 5 |
| Wed. 5/5 | EXAM. | Classification of abelian groups. Sylow theorems Extension fields of Q , finite fields. |
| Fri. 5/7 | Projects: discuss with me in my office. | . |
| Mon. 5/10 | Project Presentations. | Ben, Max. |
| Wed. 5/12 | Project Presentations. | Rubic's group: Frances, Obbie, Bobby |
| Fri. 5/14 | Project Presentations. | Amy, Barbara, Brandon, Kirsten, Maria. |