## Exercises14.5Exercises

###### 1.

Examples 14.1–14.5 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.

###### 2.

Compute all $$X_g$$ and all $$G_x$$ for each of the following permutation groups.

1. $$X= \{1, 2, 3\}\text{,}$$ $$G=S_3=\{(1), (1 \, 2), (1 \, 3), (2 \, 3), (1 \, 2 \, 3), (1 \, 3 \, 2) \}$$

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

###### 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 $$|G|=|{\mathcal O}_x| \cdot |G_x|\text{.}$$

###### 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.

1. Show that $${\mathbb R}^2$$ is a $$G$$-set.

2. Describe geometrically the orbit containing $$P\text{.}$$

3. Find the group $$G_P\text{.}$$

###### 5.

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

1. Determine the conjugacy classes (orbits) of each element of $$G\text{.}$$

2. 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.

1. $$\displaystyle S_4$$

2. $$\displaystyle D_5$$

3. $$\displaystyle {\mathbb Z}_9$$

4. $$\displaystyle Q_8$$

###### 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?

###### 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?

###### 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?

###### 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 14.28.

1. How many different compounds can be formed by replacing one or more of the hydrogen atoms with a chlorine atom?

2. 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.

###### 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 $$S_4\text{?}$$

###### 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 $$g \in G\text{,}$$ $$gC(a) g^{-1} = C(gag^{-1})\text{.}$$

###### 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 $$p^2\text{?}$$