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4. WEEK 4 1. Suppose G is a finite abelian group, a, b E G, and gcd(o(a), o(b)) = 1. (a) Prove that (a) n (b) = {eg}. (Hint. Argue that a) n (6) divides a) and (b). This can be done either using the fact that there is a bijection between subgroups of a cyclic group and positive divisors of its order, or Lagrange’s theorem.) (b) Prove that o(ab) = o(a)o(b). (Hint. suppose (ab)” = eg if and only if am = b-m. In this case, they are in (a) n (b).) (c) Prove that a € (ab). (Hint. Consider (ab)•(b).) (d) Prove that (ab) = (a,b); in particular, (a,b) is a cyclic group. 2. Suppose G is a finite group of order n, and for every positive integer m the equation a” = eg has at most m solutions in G. For every integer d, let V (d) be the number of elements of G that have order d. (a) Prove that if y(d) # 0, then \(d) = $(d) where ® is the Euler-phi function. (b) Use the fact that order of every element of g divides n to show that Edın,d>1 \(d) = n. (c) Use the previous parts and problem 5, week 1, to show that ¥(d) = $(d) if d n and d > 1. (d) Prove that G is a cyclic group. 1 2 3 4 5 6 7 8 9 10 3. Let 0 := ES10. 4 3 2 5 1 6 9 10 8 7 (a) Find a cycle decomposition of o. (b) Find out whether o is odd or even. (c) Find a cycle decomposition of o2. (d) Find o(o). (e) Find 0(T018–1), where te S10. 1. WEEK 1 1. Find all x E Z such that 3x + 7 is divisible by 11. 2. Suppose a,b, n e Z. Prove that if gcd(a, n) = gcd(b, n) = 1, then gcd(ab, n) = 1. 3. Suppose m and n are two positive integers. Prove that f : Zn + Zm, f(x)n) := [x]m is a well-defined function if and only if mln. 4. Find all the solutions of [14]21[2]21 = [28]21. 5. Let n be a positive integer. For a positive divisor d of n, let Ad:= {k e Z|15k 00(Z) = (hint. Notice that {Ad |dn, d > 0} is a partition of {1,…, n}.) = n.

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