UTK – M351 – Algebra I Spring 2007, Jochen Denzler, MWF 10:10–11:00, Ayres 205

A Zoopark of groups

Examples & Hwk 1: (ℤ, +), (ℚ, +), (ℝ, +) and (ℂ, +) are groups. — (ℕ, +) is not a group. Why not? — Is the set of irrational numbers, with +, a group? Explain.

Examples & Hwk 2: (ℤ^{∗}, ·) is not a group. Why not? — (ℚ^{∗}, ·), (ℝ^{∗}, ·) and (ℂ^{∗}, ·) are groups. Here we have used the following notations for certain sets of numbers: ℤ^{∗} := ℤ\{0}, ℚ^{∗} := ℚ\{0}, ℝ^{∗} := ℝ \ {0}, ℂ^{∗} := ℂ \ {0}.

Examples & Hwk 3: Matrix groups. We denote by GL_{n}(ℝ) the set of all invertible n × n matrices with real entries, and by SL_{n}(ℝ) the set of all those n × n matrices with real entries whose determinant is 1. By O_{n}(ℝ) we denote the set of orthogonal n × n matrices with real entries. Similarly, we write GL_{n}(ℚ) or GL_{n}(ℤ) etc. to require that the entries be rational numbers, or integers. + and · denote matrix addition or multiplication respectively. — (GL_{n}(ℝ), ·) is a group. Why? Which statement from Math 251 is pertinent here? Are SL_{n}(ℝ) and O_{n}(ℝ) groups? How about GL_{n}(ℚ), GL_{n}(ℤ), SL_{n}(ℚ), SL_{n}(ℤ), each together with ·?

Example & Hwk 4: The set {E, O} together with addition subject to the rules E + E = E, E + O = O + E = O, O + O = E is a group. Can you recognize some simple wisdom commonly known to high school graduates that is represented by this abstract example?

Example & Hwk 5: Can you generalize the previous example and construct a group on a set of three elements, let’s call it {Z, I, T }, with an appropriate addition, with a similar idea behind the abstract example?

Example & Hwk 6: The set \{a + b\sqrt{2} | a, b ∈ ℚ\} \ \{0\}, together with multiplication, is a group. Show this. Explain why it is necessary to know that \sqrt{2} is irrational in order for your proof to work.

Example & Hwk 7: The set of permutations of {1, 2, …, n}, together with composition ∘, is a group, often called S_{n}. Remember the definition of permutations: For instance, the permutation (1347256) is the function 1 ↦ 1, 2 ↦ 3, 3 ↦ 4, 4 ↦ 7, 5 ↦ 2, 6 ↦ 5, 7 ↦ 6. What is the inverse of this permutation? What is (1347256) ∘ (3512647)? What is (3512647) ∘ (1347256)?

Example & Hwk 8: Take the following set of six rational functions {I, F_{0}, F_{1}, F_{∞}, M, W } given by I(z) = z, F_{0}(z) = z/(z − 1), F_{1}(z) = 1/z, F_{∞}(z) = 1 − z, M(z) = (z − 1)/z, W (z) = 1/(1 − z). Don’t try to understand the rationale behind the names. Use composition ∘ as operation. This is a group. To help showing this, construct the group table.

Is the set of all rational functions, together with ∘, a group? Give a reason for your answer.

— Show that the set of those rational functions of the form f(z) = \frac{az+b}{cz+d} with a, b, c, d real and ad − bc ≠ 0 forms a group, again under the composition ∘ as operation.

Example & Hwk 9: Draw a regular n-gon, say, to be specific, a hexagon n = 6. Draw its six symmetry axes a_{1},…,a_{6} (half of them are lines through opposite vertices, half of them are lines through midpoints of opposite sides. Let S_{1},…,S_{6} be reflections of the hexagon with respect to the axes a_{1},…,a_{6} respectively. Moreover, let R_{0},…,R_{5} denote counterclockwise rotations about the center of the hexagon by 0, 60, 120,…,300 degrees resectively. The mappings S_{1},…,S_{6},R_{0},…,R_{5}, together with composition, form a group, which is called dihedral group D_{6}. (Or similarly D_{n}, if you have an n-gon instead of a hexagon.)

Example & Hwk 10: The set O_{2}(ℝ) × ℝ^{2} consists of pairs (A, a) where A is an orthogonal 2 × 2 matrix and a is a 2-vector. We define the multiplication by (A, a) ⊙ (B, b) = (AB, Ab + a). Check that this is a group. It is called the symmetry group of the plane, because the motiviation behind the definition of ⊙ is that (A, a) represents the mapping x ↦ Ax + a, ℝ^{2} → ℝ^{2}.