AATA Exercises

Exercises 14.5 Exercises

1.

Examples 14.1.1–14.1.5 in the first section each describe an action of a group \(G\) on a set \(X\text{,}\) which will give rise to the equivalence relation defined by \(G\)-equivalence. For each example, compute the equivalence classes of the equivalence relation, the \(G\)-equivalence classes.

Hint.

Example 14.1.1: \(0\text{,}\) \({\mathbb R}^2 \setminus \{ 0 \}\text{.}\) Example 14.1.2: \(X = \{ 1, 2, 3, 4 \}\text{.}\)

2.

Compute all \(X_g\) and all \(G_x\) for each of the following permutation groups.

\(X= \{1, 2, 3\}\text{,}\) \(G=S_3=\{(1), (1 \, 2), (1 \, 3), (2 \, 3), (1 \, 2 \, 3), (1 \, 3 \, 2) \}\)
\(X = \{1, 2, 3, 4, 5, 6\}\text{,}\) \(G = \{(1), (1 \, 2), (3 \, 4 \, 5), (3 \, 5 \, 4), (1 \, 2)(3 \, 4 \, 5), (1 \, 2)(3 \, 5 \, 4) \}\)

Hint.

(a) \(X_{(1)} = \{1, 2, 3 \}\text{,}\) \(X_{(1 \, 2)} = \{3 \}\text{,}\) \(X_{(1 \, 3)} = \{ 2 \}\text{,}\) \(X_{(2 \, 3)} = \{1 \}\text{,}\) \(X_{(1 \, 2 \, 3)} = X_{(1 \, 3 \, 2)} = \emptyset\text{.}\) \(G_1 = \{ (1), (2 \, 3) \}\text{,}\) \(G_2 = \{(1), (1 \, 3) \}\text{,}\) \(G_3 = \{ (1), (1 \, 2)\}\text{.}\)

3.

Compute the \(G\)-equivalence classes of \(X\) for each of the \(G\)-sets in Exercise 14.5.2. For each \(x \in X\) verify that \(|G|=|{\mathcal O}_x| \cdot |G_x|\text{.}\)

Hint.

(a) \({\mathcal O}_1 = {\mathcal O}_2 = {\mathcal O}_3 = \{ 1, 2, 3\}\text{.}\)

4.

Let \(G\) be the additive group of real numbers. Let the action of \(\theta \in G\) on the real plane \({\mathbb R}^2\) be given by rotating the plane counterclockwise about the origin through \(\theta\) radians. Let \(P\) be a point on the plane other than the origin.

Show that \({\mathbb R}^2\) is a \(G\)-set.
Describe geometrically the orbit containing \(P\text{.}\)
Find the group \(G_P\text{.}\)

5.

Let \(G = A_4\) and suppose that \(G\) acts on itself by conjugation; that is, \((g,h)~\mapsto~ghg^{-1}\text{.}\)

Determine the conjugacy classes (orbits) of each element of \(G\text{.}\)
Determine all of the isotropy subgroups for each element of \(G\text{.}\)

6.

Find the conjugacy classes and the class equation for each of the following groups.

\(\displaystyle S_4\)
\(\displaystyle D_5\)
\(\displaystyle {\mathbb Z}_9\)
\(\displaystyle Q_8\)

Hint.

The conjugacy classes for \(S_4\) are

\begin{gather*} {\mathcal O}_{(1)} = \{ (1) \},\\ {\mathcal O}_{(12)} = \{ (1 \, 2), (1 \, 3), (1 \, 4), (2 \, 3), (2 \, 4), (3 \, 4) \},\\ {\mathcal O}_{(1 \, 2)(3 \, 4)} = \{ (1 \, 2)(3 \, 4), (1 \, 3)(2 \, 4), (1 \, 4)(2 \, 3) \},\\ {\mathcal O}_{(123)} = \{ (1 \, 2 \, 3), (1 \, 3 \, 2), (1 \, 2 \, 4), (1 \, 4 \, 2), (1 \, 3 \, 4), (1 \, 4 \, 3), (2 \, 3 \, 4), (2 \, 4 \, 3) \},\\ {\mathcal O}_{(1234)} = \{ (1 \, 2 \, 3 \, 4), (1 \, 2 \, 4 \, 3), (1 \, 3 \, 2 \, 4), (1 \, 3 \, 4 \, 2), (1 \, 4 \, 2 \, 3), (1 \, 4 \, 3 \, 2) \}\text{.} \end{gather*}

The class equation is \(1 + 3 + 6 + 6 + 8 = 24\text{.}\)

7.

Write the class equation for \(S_5\) and for \(A_5\text{.}\)

8.

If a square remains fixed in the plane, how many different ways can the corners of the square be colored if three colors are used?

Hint.

\((3^4 + 3^1 + 3^2 + 3^1 + 3^2 + 3^2 + 3^3 + 3^3)/8 = 21\text{.}\)

9.

How many ways can the vertices of an equilateral triangle be colored using three different colors?

10.

Find the number of ways a six-sided die can be constructed if each side is marked differently with \(1, \ldots, 6\) dots.

11.

Up to a rotation, how many ways can the faces of a cube be colored with three different colors?

Hint.

The group of rigid motions of the cube can be described by the allowable permutations of the six faces and is isomorphic to \(S_4\text{.}\) There are the identity cycle, 6 permutations with the structure \((abcd)\) that correspond to the quarter turns, 3 permutations with the structure \((ab)(cd)\) that correspond to the half turns, 6 permutations with the structure \((ab)(cd)(ef)\) that correspond to rotating the cube about the centers of opposite edges, and 8 permutations with the structure \((abc)(def)\) that correspond to rotating the cube about opposite vertices.

12.

Consider \(12\) straight wires of equal lengths with their ends soldered together to form the edges of a cube. Either silver or copper wire can be used for each edge. How many different ways can the cube be constructed?

13.

Suppose that we color each of the eight corners of a cube. Using three different colors, how many ways can the corners be colored up to a rotation of the cube?

14.

Each of the faces of a regular tetrahedron can be painted either red or white. Up to a rotation, how many different ways can the tetrahedron be painted?

15.

Suppose that the vertices of a regular hexagon are to be colored either red or white. How many ways can this be done up to a symmetry of the hexagon?

Hint.

\((1 \cdot 2^6 + 3 \cdot 2^4 + 4 \cdot 2^3 + 2 \cdot 2^2 + 2 \cdot 2^1)/12 = 13\text{.}\)

16.

A molecule of benzene is made up of six carbon atoms and six hydrogen atoms, linked together in a hexagonal shape as in Figure 14.5.1.

How many different compounds can be formed by replacing one or more of the hydrogen atoms with a chlorine atom?
Find the number of different chemical compounds that can be formed by replacing three of the six hydrogen atoms in a benzene ring with a \(CH_3\) radical.

A hexagon with hydrogen atoms, H, attached at each vertex. — Figure 14.5.1. A benzene ring

17.

How many equivalence classes of switching functions are there if the input variables \(x_1\text{,}\) \(x_2\text{,}\) and \(x_3\) can be permuted by any permutation in \(S_3\text{?}\) What if the input variables \(x_1\text{,}\) \(x_2\text{,}\) \(x_3\text{,}\) and \(x_4\) can be permuted by any permutation in \(S_4\text{?}\)

Hint.

\((1 \cdot 2^8 + 3 \cdot 2^6 + 2 \cdot 2^4)/6 = 80\text{.}\)

18.

How many equivalence classes of switching functions are there if the input variables \(x_1\text{,}\) \(x_2\text{,}\) \(x_3\text{,}\) and \(x_4\) can be permuted by any permutation in the subgroup of \(S_4\) generated by the permutation \((x_1, x_2, x_3, x_4)\text{?}\)

19.

A striped necktie has \(12\) bands of color. Each band can be colored by one of four possible colors. How many possible different-colored neckties are there?

20.

A group acts faithfully on a \(G\)-set \(X\) if the identity is the only element of \(G\) that leaves every element of \(X\) fixed. Show that \(G\) acts faithfully on \(X\) if and only if no two distinct elements of \(G\) have the same action on each element of \(X\text{.}\)

21.

Let \(p\) be prime. Show that the number of different abelian groups of order \(p^n\) (up to isomorphism) is the same as the number of conjugacy classes in \(S_n\text{.}\)

22.

Let \(a \in G\text{.}\) Show that for any \(g \in G\text{,}\) \(gC(a) g^{-1} = C(gag^{-1})\text{.}\)

Hint.

Use the fact that \(x \in g C(a) g^{-1}\) if and only if \(g^{-1}x g \in C(a)\text{.}\)

23.

Let \(|G| = p^n\) be a nonabelian group for \(p\) prime. Prove that \(|Z(G)| \lt p^{n - 1}\text{.}\)

24.

Let \(G\) be a group with order \(p^n\) where \(p\) is prime and \(X\) a finite \(G\)-set. If \(X_G = \{ x \in X : gx = x \text{ for all }g \in G \}\) is the set of elements in \(X\) fixed by the group action, then prove that \(|X| \equiv |X_G| \pmod{ p}\text{.}\)

25.

If \(G\) is a group of order \(p^n\text{,}\) where \(p\) is prime and \(n \geq 2\text{,}\) show that \(G\) must have a proper subgroup of order \(p\text{.}\) If \(n \geq 3\text{,}\) is it true that \(G\) will have a proper subgroup of order \(p^2\text{?}\)

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