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Mathematics 466 A. Hulpke Homework (due Nov. 7) 45) Let G be a (not necessarily abelian!) group and a, b ∈ G. Prove that ∣ab∣ = ∣ba∣. 46) a) Determine a cyclic and a non-cyclic subgroup of U(40). b) Is U(8) a subgroups of U(32)? Explain! 47) Let G = ⟨(1, 2, 3, 8)(4, 5, 6, 7), (1, 7, 3, 5)(2, 6, 8, 4)⟩ be a group generated by two permutations. a) Determine the elements of G and their orders. (GAP can help here!) What is ∣G∣? b) Determine all subgroups of G. (Hint: Every proper subgroup has at most 4 elements. If it contains an element of order 4 it therefore must be cyclic.) c) Show that all subgroups of G are normal, but that G is not abelian. 48) (GAP) Consider a puzzle, as depicted on the side, which consists of two overlapping rings filled with balls. By moving the balls along either ring one can mix the balls and then try to restore the original situation. In this problem we will consider a simplified version, that consists only of 8 balls, arranged as given in the second picture on the right. We will describe this state by a scheme of the form: Image: Egner,Püschel: Proc ISSAC 1998 1 1 3 2 4 2 5 4 7 6 8 6 3 5 7 8 a) Write permutations for the rotation λ around the left circle and the rotation ρ around the right circle. b) Using the Factorization command, give sequences of operations that will return the following states back to the original: 4 5 2 1 1 2 3 8 4 , 6 3 5 7 7 6 8 (Check that the results you got actually work!) 49) Let G = S n and a i = (i, i + 1) for i = 1, . . . , n − 1. a) Let g ∈ S n with and g(n) = i. Show that b = g ⋅ a i ⋅ a i+1 ⋅ ⋯ ⋅ a n−1 fulfills that b(n) = n, i.e. that b ∈ S n−1 . b) Using the argument from a), show by induction over n that S n = ⟨a1 , . . . , a n−1 ⟩. c) Show that a i a j = a j a i if ∣i − j∣ ≥ 2 and a i a j a i = a j a i a j if ∣i − j∣ = 1. (Note: These relations are called “braid relations”, the name comes from representing the braiding of strings. As shown in the picture on the side, relations of this type hold for the “twists” of adjacent strings. One can show that using these braid relations as well as relations a 2i = e, one can bring every word in S n into a shortest form.) a1: Move left string over middle one, a2: Move middle string over right one, a1 a2 a2 a1 a1 a2 50) Let G be a group and H, K ≤ G with H ≤ K ≤ G such that [G ∶ H] is finite. a) Let {r i } a set of representatives of the cosets of K in G and {s j } a set of representatives of the cosets of H in K. Show that the products {s j ⋅ r i } form a set of representatives for the cosets of H in G. (In other words: Show that for every x ∈ G there is a pair r, s such that Hx = (s j r i ), and that H(s j r i ) = H(s y r x ) only if i = x and j = y. Hint: It is easiest to first consider cosets of K to take care of the r-part.) b) Show that [G ∶ H] = [G ∶ K] ⋅ [K ∶ H]. (You cannot use Lagranges theorem, as we don’t assume that G is finite!) Problems marked with a ∗ are bonus problems for extra credit.