MATH 4010/6010 Homework

Problem Set 6, due Thursday, March 5

Problems to work but not turn in:
7.2: 4, 6
7.3: 1, 2, 3

Problems to turn in:
A. Prove that the conjugacy class of S₅ consisting of the 24 cycles of length 5 splits into two conjugacy classes in A₅.
7.2: 10
7.3: 5, 6, 10, 14

Challenge problems (extra credit, required for MATH 6010):

B. Prove that the group G of rotation symmetries of an icosahedron is isomorphic to the alternating group A₅, by considering the action of G on five tetrahedra in the icosahedron. (Cf. problem 8 in section 7.2 of the textbook.)

C. If any one of k different symbols may be placed at each vertex of a regular hexagon, then the number of distinct patterns formed (allowing for rotations and reflections) is (k⁶ + 3k⁴ + 4k³ + 2k² + 2k)/12.

Problem C comes from chemistry. The benzene molecule C₆H₆ is a regular hexagon with a carbon atom C at each vertex and a hydrogen atom H attached to each C. Other atoms such as F (flourine) or Cl (chlorine) can be substituted for the hydrogen atoms. How many different molecules can you make if you can substitute k different types of atoms for the hydrogen atoms? (Hydrogen counts as one of the k types.)

Problem Set 7, due Thursday, March 26 (not March 19 as previously announced)

Problems to work but not turn in:
7.5: 3, 4, 6, 7

Problems to turn in:
7.5: 9, 13, 14, 17, 20, 23

Challenge problems (extra credit, required for MATH 6010):
7.5: 24, 27

Problem Set 8, due Thursday, April 2

Problems to work but not turn in:
5.1: 4, 11, 13, 15 (assigned in MATH 4000/6000)

Problems to turn in:
5.1: 16, 18
7.6: 1bce, 2, 3, 4

Challenge problems (extra credit, required for MATH 6010):
7.6: 20, 21