数学代写 - Mathematics 322 — Term Test
时间:2020-11-20
Question 1.[4 marks] Determine whether or not the permutations σ = (1234)(256)(23) and (3412)(134)(16) are conjugate in S6. Question 2. Consider the group G = S6. For each i, let Hi be the subgroup of G fixing the element i. Let X = {H1, . . . , H6}. a) [3 marks] Show that if 1 ≤ i ≤ 6 and g ∈ G, then gHig 1 = Hj for some j depending on i and g, and that the function φ(g) : Hi 7→ Hj is a permutation of the set X. b) [2 marks] Show that g 7→ φ(g) is a homomorphism G → SX, where SX is the group of permutations of X. c) [6 marks] Show that the action defined by the homomorphism φ in part (b) is both transitive and faithful. For each Hi , find the stabilizer in G of Hi.
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