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Mathematics 1214: Introduction to Group Theory Tutorial exercise sheet 7 1. Let n ∈ N, and let α : Z → Z be the mapping α(m) = m + 1 for m ∈ Z. As usual, we write αk = |α ◦ ·{z · · ◦ α} if k ∈ N, α0 = ιZ and we also define αk = n times (α−1 )−k if k ∈ Z with k < 0. (a) Show that αk (m) = m + k for m, k ∈ Z. (b) Consider the set G = {αnd : d ∈ Z}. [In this expression for G, the variable n is fixed, but d is a dummy variable running over Z. Check that you understand what this means!] Show that G is a permutation group on Z. (c) Show that G-orbit equivalence is the same as congruence modulo n. 2. Let a, b, q, r be integers such that a = bq + r. Show that gcd(a, b) = gcd(b, r). 3. It may be shown that if a, b are any integers, then there exist integers s, t such that as + bt = gcd(a, b). Use this to prove that if n ∈ N and c, d ∈ Z with gcd(c, n) = 1 then there is a solution x ∈ Z to the congruence cx ≡ d (mod n). 4. Suppose that n ∈ N and k ∈ Z with n k. Prove that for m ∈ Z we have n m ⇐⇒ n m + k. 5. (a) Show that for any real number x and any j ∈ N, we have xj − 1 = (x − 1)(xj−1 + xj−2 + · · · + x2 + x + 1). (b) Deduce that if b ∈ Z with b ≥ 2 and j ∈ N, then bj ≡ 1 (mod b − 1). (c) Show that if n ∈ N0 and a1 , a2 , . . . , an ∈ Z, then b−1 n X j=0 j aj b ⇐⇒ b − 1 n X aj . j=0 (d) Is 123456789123456789123456789123456789 divisible by 9? [Hint: take b = 10 in part (c)].