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Exercises 7.6 Exercises

1.

Examples 7.5.1–7.5.5 in Section 7.5 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 7.5.1: \(0\text{,}\) \({\mathbb R}^2 \setminus \{ 0 \}\text{.}\) Example 7.5.2: \(X = \{ 1, 2, 3, 4 \}\text{.}\)

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. \(X = \{1, 2, 3, 4, 5, 6\}\text{,}\) \(G = \{(1), (1 \, 2), (3 \, 4 \, 5), (3 \, 5 \, 4), (1 \, 2)(3 \, 4 \, 5), (1 \, 2)(3 \, 5 \, 4) \}\)
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 7.6.2. 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.
  1. Show that \({\mathbb R}^2\) is a \(G\)-set.
  2. Describe geometrically the orbit containing \(P\text{.}\)
  3. Find the group \(G_P\text{.}\)
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