Homework

HW 13 Solution:

6.2 See the link (both proves are similar): http://groupprops.subwiki.org/wiki/Borel_subgroup_is_self-normalizing_in_general_linear_group

7.3 There is the unique Sylow 5-subgroup by Sylow's theorem. So there are 4 elements of order 5.

7.5 (a) D_10 is isomorphic to the product of D_5 and C_2 (cyclic group of order 2). There are 5 Sylow 2-subgroup in D_5. Thus there are 5 Sylow 2-subgroups of order 4 in D_10.

(b) The class equation for T is 12=1+3+4+4, where the later two conjugacy classes contain elements of order 3. So the Sylow 2-subgroup of order 4 is unique, containing rotations along edges.

(c) The class equation for O is 24=1+3+6+6+8. By Sylow's third theorem, there are either 1 or 3 Sylow 2-subgroup of order 8. First it is easy to construct one 2-subgroup H, which contains the rotations along a face plus rotations along edges perpendicular to that face. It is easy to see H is isomorphic to D_4. There are three pairs of opposite faces so there are 3 Sylow 2-subgroups.

(d) The class equation for I is 60= 1+20+12+12+15. The middle 3 conjugacy classes are irrelevant because they consists elements of order 3 or 5. First we construct a Sylow 2 subgroup of order 4. Given a rotation along an edge E, by geometry, there exist two different rotations along edges E' and E" commuting with the given rotation. They forms a Klein 4 -group. There are 15 rotations along edges. So there are 5 Sylow 2-subgroups.

7.7 See P207 proof of the third Sylow theorem. There are two H-orbits: H itself and the other orbit consisting p Sylow p-subgroups.

Tips: I recommend that if you want to learn the course well, you should do the problems in the book as many as possible (ideally all of them). I am happy to talk to you during the office hour if you need extra help.

Due Dec 13 (Friday) - Ch.7 Ex. 6.2, 7.3, 7.5, 7.7

Due Dec 9 (Monday) - Ch.7 Ex. 2.2, 2.8, 2.14, 3.1, 3.2, 4.4, 4.5, 4.6

Due Nov 27 (Wednesday) - Ch. 6 Ex. 8.3, 9.1, 9.2, 11.3, 11.6, 12.3.

Due Nov 22 (Friday) - Ch. 6 Ex. 4.1, 4.2(a), 4.3, 7.1, 7.8, 7.9.

Due Nov 8 (Friday) - Ch. 4 Ex. 4.3, 4.6, 5.6, 6.2, 6.3, 6.4, 7.3, Ch. 5 Ex. 1.1, 1.3

Previous Homework

Due Sep 4 - Ch.2: Ex. 1.1, 1.2, 2.2, 2.3, 2.4, 2.5

Due Sep 11 - Ch.2: Ex. 3.2, 4.1, 4.2, 4.5, 4.6, 4.10

Due Sep 18 - Ch.2: Ex. 5.2, 5.3, 6.1, 6.2, 6.8, 6.10

Due Sep 25 - Ch.2: Ex. 8.3, 8.6, 8.7, 9.1, 10.3, 10.4

Due Oct 2 - Ch.2: Ex. 11.2, 11.4, 11.6, 11.7, 12.1, 12.5

Due Oct 11 - Ch.3: Ex. 2.2, 2.3, 2.4, 2.8, 2.9, 3.2 (aka* Ex. 1.2, 1.3, 1.4, 1.8, 1.9, 2.2)

Due Oct 23 - Ch.3: Ex. 4.1, 4.2, 4.5, 4.8. 5.1, 5.2 (aka* Ex. 3.1, 3.2, 3.5, 3.8, 4.1, 4.2)

Due Oct 30 - Ch.4: Ex. 1.3, 1.5, 2.1, 2.4, 3.2, 3.3

*it appears that the second edition of the book was first printed with all the numbering of the sections in the exercises of chapter 3 off by 1 (starting with "Section 1 Fields" - note that in the chapter the section entitled "Fields" is section 2). This was amended in a reprint of the same (second) edition.