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Burnside Counting, Isometry

1) Let E1, E2, E3, and E4 denote, respectively, the edges ab, bc, cd, and da of the square in the figure below. Write the permutation induced on {E1, E2, E3, E4} by each isometry (symmetry) of the square. [Example: ρH ↦ (E1E3).] Does the symmetry group of the square act faithfully on {E1, E2, E3, E4}

2) The group G = 〈(1 2 3 4)(5 6)〉 is of order 4 and acts on {1, 2, 3, 4, 5, 6}.
a. Determine Orb(k) for 1 ≤ k ≤ 6.
b. Determine Gk for 1 ≤ k ≤ 6.
c. Use parts (a) and (b) to verify that |Orb(k)| = |G|/|Gk| for 1 ≤ k ≤ 6.

3) Let S denote the collection of all subgroups of a finite group G. For a ϵ G and H ϵS, let πa(H) = aHa?1.
a. Verify that with this definition G acts on S. (Each subgroup aHa?1 is called a conjugate of H.)
b. For G = S3, determine Orb(〈(1 2)〉).
c. For G = S3, determine G〈(1 2)〉.
d. Use parts (b) and (c) to verify that
e. For H ϵ S, the normalizer of H in G is defined by NG(H) = {a ϵ G :aHa?1 =H}. Using results from this section, explain why NG(H) is a subgroup ofG, and the number of conjugates of H is [G : NG(H)].

4) Prove that if a finite group G contains a subgroup H ≠ G such that |G| ∤ [G :H]!, then H contains a nontrivial normal subgroup of G. (note use Lagrange?s Theorem, and facts about homomorphisms.)

5) Use Burnside?s Counting Theorem to compute the number of orbits for the group 〈(1 2 3 4)(5 6)〉 acting on {1, 2, 3, 4, 5, 6}. What are the orbits?
6) Consider the problem of painting the edges of a square so that one is red, one is white, one is blue, and one is yellow.
a. In how many distinguishable ways can this be done if the edges of the square are distinguishable?
b. Repeat (a), except count different ways as being indistinguishable if one can be obtained from the other by rotation of the square in the plane.
c. Repeat (b), except permit reflections through lines as well as rotations in the plane.


Solution Preview

Please see the attached file.

Identity ↦ identity
Rotation around π/2 ↦ (E_1 〖,E〗_2 〖,E〗_(3,) E_4)
Rotation around π ↦ (E_1 〖,E〗_3)(E_(2,) E_4)
Rotation around 3π/2 ↦ (E_1 〖,E〗_4 〖,E〗_3,E_2)
Symmetry along the line (Vp) ↦ (E_2,E_4)
Symmetry along the line (Hp) ↦ (E_1 〖,E〗_(3,))
Symmetry along the line (ac) ↦ (E_1 E_4)( E_2 〖,E〗_(3,))
Symmetry along the line (bd) ↦ (E_1 〖,E〗_2)〖,(E〗_(3,) E_4)

Symmetry group acts faithfully on {E_1 〖,E〗_2 〖,E〗_(3,) E_4} , since only the identity symmetry is mapped to the identity by this mapping.

From the problem 5 we know that there are two orbits.
a)The orbit of 1 is {1,2,3,4,}.
The orbit of 2 is {1,2,3,4}.
The orbit of 3 is {1,2,3,4}.
The orbit of 4 is {1,2,3,4}.
The orbit of 5 is {5,6,}.
The orbit of 6 is {5,6}.
b) G_1=G_2=G_3=G_4={1} and G_5 〖=G〗_6={1,a^2}
c) For 1≤k≤4, |Orb(k)|=4, |G_k|=1. And we have 4=4/1.
For 1≤k≤4, |Orb(k)|=2, 〖|G〗_k ||=2. We have 2=4/2.
3. a) Let us first verify that aHa^(-1) is also a subgroup. Indeed, ...

Solution Summary

Burnside counting and isometry is examined. The expert determines whether the symmetry group of the square act faithfully is determined.