Homework for Week 8

Part 1

  1. (Rotman 1.37) True or false, with reasons.
    1. If a finite group G acts on a set X then X must be finite.
    2. If a group acts on a finite set X then G must be finite.
    3. If a group G acts on a set X and if x,y are in X then Stab(x) is isomorphic to Stab(y).
    4. If a group G acts on a set X, and if x,y are both in the same orbit in X, then Stab(x) is isomorphic to Stab(y).
  2. Suppose that G is a group and X=G is the same group.
    1. Prove that if g is in G and x is in X then the action defined by g(x) = gxg-1 is a group action.
    2. If G=X=S5 and g is the permutation (13), what does the function g(x) do to each permutation x in X?
  3. (Rotman 1.24, parts) Try to relate this to group actions!
    1. How many permutations in S5 commute with (12)(34)?
    2. How many permutations in S7 commute with (12)(345)?
    3. List all of the permutations in S7 that commute with (12)(345).
  4. How many rotational and reflectional symmetries of the pentagon are there?
  5. How many rotational symmetries does a cube have?

Part 2

  1. How many rotational symmetries does a dodecahedron have?
  2. (Artin, Chapter 5, Problem 5.5) Let G be the dihedral group of symmetries of the square (meaning the eight rotational and reflectional symmetries). For the purposes of this problem, an edge is considered stabilized if it's sent to itself regardless of what direction it's drawn.
    1. What is the stabilizer of a vertex of the square? Of an edge of the square?
    2. G acts on the set of diagonal lines. How many elements are in the set of diagonal lines? What is the stabilizer of a diagonal?
  3. (Tucker, Applied Combinatorics, 9.2.1) How many different 4-bead necklaces are there using beads of red, white, blue, and green (assume necklaces can rotate but cannot flip over)?
  4. (Tucker, Applied Combinatorics, 9.2.5) A merry-go-round can be built with three different styles of horses. How many five-horse merry-go-rounds are there?
  5. (Modified from Tucker, Applied Combinatorics, 9.3) How many ways are there to 3-color the corners of a pentagon that are:
    1. Distinct with respect to rotations only?
    2. Distinct with respect to rotations and reflections?
    3. Find two 3-colorings of the pentagon that are different in the first part but equivalent in the second.