2. Consider the subgroups C4 and D4 of O(2). (a) Write down the orbit of the point p = (1, 1) ? ?² under each of the C4 and D4 actions. State the orbit-stabilizer theorem and explain how these orbits illustrate this. [8 marks]
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[(2+2)+(2+2)+2=10 Marks] Let G = {e, (123), (132), (45), (123)(45), (132)(45)}, a subgroup of S5. Then G acts on the set X = {1, 2, 3, 4, 5} in a natural way by the rule σ · n = σ(n) ∀ σ ∈ G, n ∈ X [e.g., if σ = (132), then σ · 2 = 1 and σ · 5 = 5]. (a) List all the distinct elements of the following orbits: (i) The orbit of 1, O1. (ii) The orbit of 4, O4. (b) List all the distinct elements of the following stabilizers: (i) The stabilizer of 1, G1. (ii) The stabilizer of 4, G4. (d) Verify the Orbit-Stabilizer Theorem in the case of the element 4 of X, that is, show that |O4| = [G : G4].
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