Web page of Dmytro Savchuk |
Homework problems will be listed here
| # | Due on | Problems to solve | Problems to submit |
|---|---|---|---|
| 1 | R 9/3 |
Section 1.1: 2, 4, 6, 7, 8, 12, 13, 14, 15 Section 1.2: 1, 3, 4, 6, 9, 10, 11, 12, 19 |
Section 1.1: 12, 15 Section 1.2: 3, 11 |
| 2 | R 9/10 |
Section 1.3: 1, 5, 8, 9 Section 1.4: 2, 3, 4, 5, 6, 8 Section 1.5: 1, 5, 6, 7, 8, 9, 14, 15 |
Section 1.3: 5, 8 Section 1.4: 3, 6 Section 1.5: 9, 14 |
| 3 | R 9/17 |
Section 1.5: 19 - A group is called residually finite if there is a sequence of finite index subgroups of this group with the trivial intersection. Prove that the group Q of all rational numbers does not have proper finite index subgroups and, thus, is not residually finite. Section 1.6: 1, 3, 6, 7, 8, 10, 12 - Draw the Cayley graph of S3 with respect to the generating set {(1,2),(1,2,3)} Section 1.7: 1, 2, 7 |
- A group is called residually finite if there is a sequence of subgroups of this group
with the trivial intersection. Prove that the group Q of all rational numbers does not have
finite index subgroups and, thus, is not residually finite. Section 1.6: 6, 8 - Draw the Cayley graph of S3 with respect to the generating set {(1,2),(1,2,3)} Section 1.7: 7 |
| 4 | T 9/29 |
Section 1.8: 2, 5, 6, 9 Section 1.9: 1, 3, 5, 6, 8, 11, 12 Section 2.1: 2, 3, 5, 7, 9, 10 |
Section 1.8: 6 Section 1.9: 6, 12 Section 2.1: 7, 10 |
| 5 | R 10/8 |
Section 2.2: 1, 3, 5, 7, 12, 13 |
Section 2.2: 1, 7, 12 |
| 6 | F 10/16 |
Section 2.4: 1, 2, 4, 6, 7, 8, 9, 13 Problem A on Burnside's Lemma: What is the number of rotationally distinct colorings of the faces of a tetrahedron using three colours? Section 2.5: 1, 6, 7, 9, 12, 13 |
Section 2.4: 7, 13, Problem A Section 2.5: 9, 13 |
| 7 | F 10/23 |
Section 2.6: 1, 3 (use one of the proposition in the section), 4 Section 2.7: 1, 2, 4, 5, 6, 7, 9, 10, 11, 14 Section 2.8: 1, 2, 3, 5, 6, 7, 11 |
Section 2.6: 1 Section 2.7: 4, 11 Section 2.8: 3, 7 |
| 8 | F 10/30 |
Section 3.1: 1, 2, 3, 6, 9, 11, 12, 14, 18 Problem B related to Kaplansky's zero-divizors conjecture: Let R be a ring and G be a group with torsion. Prove that the group ring R[G] has zero divisors. Section 3.2: 1, 2, 3, 4, 7, 10 |
Section 3.1: 6, 14, Problem B Section 3.2: 7, 10 |
| 9 | F 11/6 |
Section 3.2: 11, 13, 18, 19, 23, 25 Section 3.3: 1, 2, 6, 7, 10, 11 |
Section 3.2: 19, 23 Section 3.3: 3, 7, 11 |
| 10 | F 11/6 |
Section 3.4: 1, 2, 3, 5, 6, 9, 11, 12, 13 Section 3.5: 1, 2, 3, 6, 7, 8 |
This assignment will not be collected |
| 11 | T 11/24 |
Section 3.6: 1, 3, 4, 5, 7, 9, 10, 11, 12 Section 4.1: 1, 2, 3, 5, 6, 7, 8, 12, 15 |
Section 3.6: 5, 9 Section 4.1: 6, 12, 15 |
| 12 | F 12/4 |
Section 4.2: 1, 6, 8, 9, 12, 13 Section 4.3: 1, 3, 4, 5, 7 |
Section 4.2: 1, 9, 13 Section 4.3: 5, 7 |
| 13 | T 12/10 |
Section 4.6: 1, 2, 3, 4, 6 |
This assignment will not be collected |
Updated:
December 03rd, 2015
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