91Ó°ÊÓ

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

Let \(G=A_{4}\) and suppose that \(G\) acts on itself by conjugation; that is, \((g, h) \mapsto g h g^{-1}\). (a) Determine the conjugacy classes (orbits) of each element of \(G\). (b) Determine all of the isotropy subgroups for each element of \(G\).

Find the conjugacy classes and the class equation for each of the following groups. (a) \(S_{4}\) (b) \(D_{5}\) (c) \(\mathbb{Z}_{9}\) (d) \(Q_{8}\)

Write the class equation for \(S_{5}\) and for \(A_{5}\).

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?

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

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

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?

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?

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 . (a) How many different compounds can be formed by replacing one or more of the hydrogen atoms with a chlorine atom? (b) 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 \(C H_{3}\) radical.