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Section Supplementary Problems

1

Show that a function from \(S\) to \(T\) has an inverse (defined on \(T\)) if and only if it is a bijection.

2

How many elements are in the dihedral group \(D_3\text{?}\) The symmetric group \(S_3\text{?}\) What can you conclude about \(D_3\) and \(S_3\text{?}\)

3

A tetrahedron is a thee dimensional geometric figure with four vertices, six edges, and four triangular faces. Suppose we start with a tetrahedron in space and consider the set of all permutations of the vertices of the tetrahedron that correspond to moving the tetrahedron in space and returning it to its original location, perhaps with the vertices in different places.

  1. Explain why these permutations form a group.
  2. What is the size of this group?
  3. Write down in two-row notation a permutation that is not in this group.
4

Find a three-element subgroup of the group \(S_3\text{.}\) Can you find a different three-element subgroup of \(S_3\text{?}\)

5

Prove true or demonstrate false with a counterexample: “In a permutation group, \((\sigma\varphi)^n = \sigma^n\varphi^n\text{.}\)”

6

If a group \(G\) acts on a set \(S\text{,}\) and if \(\sigma(x) =y\text{,}\) is there anything interesting we can say about the subgroups \(\Fix(x)\) and \(\Fix(y)\text{?}\)

7

  1. If a group \(G\) acts on a set \(S\text{,}\) does \(\overline{\sigma}(f) = f\circ\sigma\) define a group action on the functions from \(S\) to a set \(T\text{?}\) Why or why not?

  2. If a group \(G\) acts on a set \(S\text{,}\) does \(\sigma(f)=f\circ\sigma^{-1}\) define a group action on the functions from \(S\) to a set \(T\) ? Why or why not?

  3. Is either of the possible group actions essentially the same as the action we described on colorings of a set, or is that an entirely different action?

8

Find the number of ways to color the faces of a tetrahedron with two colors.

9

Find the number of ways to color the faces of a tetrahedron with four colors so that each color is used.

10

Find the cycle index of the group of spatial symmetries of the tetrahedron acting on the vertices. Find the cycle index for the same group acting on the faces.

11

Find the generating function for the number of ways to color the faces of the tetrahedron with red, blue, green and yellow.

12

Find the generating function for the number of ways to color the faces of a cube with four colors so that all four colors are used.

13

How many different graphs are there on six vertices with seven edges?

14

Show that if \(H\) is a subgroup of the group \(G\text{,}\) then \(H\) acts on \(G\) by \(\sigma(\tau) = \sigma\circ\tau\) for all \(\sigma\) in \(H\) and \(\tau\) in \(G\text{.}\) What is the size of an orbit of this action? How does the size of a subgroup of a group relate to the size of the group?