Math 502 schedule
| Tuesday |
Thursday |
August 27
In Lecture
- Definition of a group, basic examples
- Permutation groups
- The origin of group theory in Galois Theory
Associated Reading:
Important Note: There will be no recitations this week.
| August 29
Lecture
- Subgroups, homomorphisms.
- Cosets, normal subgroups, quotient groups, exact sequences
- Symmetric groups
Associated Reading:
|
September 3
In Lecture
- Operations of groups on sets
- Matrix groups
Associated Reading:
| September 5
Lecture
- Error correction and subgroups of (Z/2)^n
- Isometry groups
Associated Reading:
|
September 10
In Lecture
- The orbit formula for groups acting on sets
- Calculations in symmetric groups
- The class equation
Associated Reading:
- D-F: 1.3, 2.2, 4.1, 4.2, 4.3, 4.4, 4.5
| September 12
Lecture
- Consequences of the class equation
- p-groups have non-trivial centers
- Statement of Sylow theorems
Associated Reading:
|
September 17
In Lecture
- Examples of pplications of the Sylow theorems
Associated Reading:
- D-F: 1.2, 1.3, 2.5, 3.4, 4.3, 4.5
| September 19
Lecture
- Examples of the Sylow theorems, continued
- Semi-direct products
Associated Reading:
- D-F: 1.2, 1.3, 2.5, 3.4, 4.3, 4.5
|
September 24
In Lecture
- Proof of the Sylow theorems
- Simple groups of order 60
Associated Reading:
- D-F: 1.2, 1.3, 2.5, 3.4, 4.3, 4.5
| September 26
Lecture
- Simple groups of order 60, continued
Associated Reading:
- D-F: 2.5, 3.4, 4.3, 4.4, 4.5
|
October 1
In Lecture
- End of discussion of simple groups of order 60
- Review for the midterm
Associated Reading:
- D-F: 1.2, 1.3, 2.5, 3.4, 4.3, 4.5, 5.5
Suggested review problems from D-F:
- Section 1.2, number 9.
- Section 1.3, number 15.
- Section 1.6, number 26.
- Section 1.7, number 8.
- Section 3.1, number 33.
- Section 3.2, number 21.
- Section 4.1, number 6.
- Section 4.3, number 13.
- Section 4.4, number 7.
- Section 4.5, numbers 26, 30, 45.
- Show that the automorphism group of (Z/5) \times (Z/5)
is GL_2(Z/5), and use this to construct a
non-abelian
group of order 75.
Use this to do problem 8 of section 5.5
| October 3
Lecture
|
October 8
Lecture
- Schreier's Theorem
- Jordan Holder Theorem
- Classifying finite simple groups
Associated Reading:
- D-F: 2.5, 3.4, 4.3, 4.4, 4.5
| October 10
No class - fall break!
|
October 15
Lecture
- Schreier's Theorem and the Butterfly lemma
- Finite abelian groups
Associated Reading:
| October 17
Lecture
- Solvable and Nilpotent groups
- Derived series, upper central series, lower central series
Associated Reading:
|
October 22
Lecture
- Derived series, upper central series and lower central series
Associated Reading:
| October 24
Lecture
Associated Reading:
|
October 29
Lecture
- Group extensions, continued
- Crystallographic groups
Associated Reading:
| October 31
Lecture
Associated Reading:
Suggested review problems from D-F for the mid-term on Nov. 7.:
- Section 3.4, number 2
- Section 3.4, number 8.
- Section 5.2, number 1.
- Section 5.2, number 11
(The rank of a finite abelian group G is the minimal
number of cyclic groups whose product is isomorphic to G.)
- Section 6.1, number 1.
- Section 6.1, number 6.
- Section 6.1, number 9.
- Section 6.1, number 23.
- Section 6.1, number 18.
- Section 17.4, number 1.
- Identify R^2 with the complex numbers C. In class we talked about
the group G of isometries generated by a(z) = 2 + \overline{z} and
b(z) = z + 2*\sqrt(-1), where \overline{z} is the complex conjugate of z.
Show that when T_2 is the group of all translations, G \cap T_2 has
index 2 in G. Then show that G is not a semi-direct product of G\cap T_2
with a group of order 2 contained in G.
|
Nov. 5
Lecture
Associated Reading:
- Review all the readings since October 8, inclusive
| Nov. 7
Lecture
Suggested review problems from D-F for the mid-term:
- Section 3.4, number 2
- Section 3.4, number 8.
- Section 5.2, number 1.
- Section 5.2, number 11
(The rank of a finite abelian group G is the minimal
number of cyclic groups whose product is isomorphic to G.)
- Section 6.1, number 1.
- Section 6.1, number 6.
- Section 6.1, number 9.
- Section 6.1, number 23.
- Section 6.1, number 18.
- Section 17.4, number 1.
- Identify R^2 with the complex numbers C. In class we talked about
the group G of isometries generated by a(z) = 2 + \overline{z} and
b(z) = z + 2*\sqrt(-1), where \overline{z} is the complex conjugate of z.
Show that when T_2 is the group of all translations, G \cap T_2 has
index 2 in G. Then show that G is not a semi-direct product of G\cap T_2
with a group of order 2 contained in G.
|
Nov. 12
Lecture
- Integral domains, division rings, fields
- Ideals, quotient rings
Associated Reading:
- D-F: 7.1 - 7.4 and Appendix 1
| Nov. 14
Lecture
- Maximal ideals and prime ideals.
- Zorn's Lemma and applications
Associated Reading:
- D-F: 7.1 - 7.4 and Appendix 1, 15.5
|
Nov. 19
Lecture
- Prime ideals, Spec(R) and the Zariski topology
Associated Reading:
| Nov. 21
Lecture
- The Zariski topology, continued
Associated Reading:
|
Nov. 26
Lecture
Associated Reading:
| Nov. 28
Lecture
|
Dec. 3
In Lecture:
Associated Reading:
- Dummit and Foote, section 8.1
| Dec. 5
In Lecture:
- Chinese Remainder theorem
- Review of the semester
Associated Reading:
Final exam information:
The final exam will be from 12 p.m. to 2 p.m. on Wednesday, Dec. 18, in
room A2 of DRL labs. You can bring one two-sided handwritten
page of notes to the exam.
Here are some review problems from D-F for the final exam.
These have to do with material since the second mid-term.
See the earlier suggested problem lists for prior review material.
- Section 7.1, numbers 20, 26
- Section 7.2, numbers 5. 12.
- Section 7.3, numbers 10, 15.
- Section 7.4, numbers 1, 8, 16, 23, 41.
Section 7.5, numbers 5, 6.
Section 7.6, numbers 1, 2.
Section 8.1, number 2(c), 7 (only for the first example), 12.
Section 15.5, numbers 1, 2 9
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