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## Section14.4Exercises

###### 1

Examples 14.114.5 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. $X= \{1, 2, 3\}\text{,}$ $G=S_3=\{(1), (12), (13), (23), (123), (132) \}$

2. $X = \{1, 2, 3, 4, 5, 6\}\text{,}$ $G = \{(1), (12), (345), (354), (12)(345), (12)(354) \}$

###### 3

Compute the $G$-equivalence classes of $X$ for each of the $G$-sets in Exercise 14.4.2. 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. $S_4$

2. $D_5$

3. ${\mathbb Z}_9$

4. $Q_8$

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