# hw3.pdf - MODERN ALGEBRA 1 HOMEWORK 3(1 Chapter 2 4.1(2 In...

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MODERN ALGEBRA 1: HOMEWORK 3 (1) Chapter 2: 4.1 (2) In the following problem, G is a group. All small letters stand for elements of G . (a) Show that h x i = h x - 1 i . Conclude that x and x - 1 have the same order. (b) Show that x and yxy - 1 have the same order. Use this to show that ab and ba have the same order. (c) Let H be a group and φ : G H an isomorphism. Show that x and φ ( x ) have the same order. (3) Let G be a cyclic group of order n , and let x G be a generator. Let H G be a subgroup. By considering the set S Z defined by S = { i Z | x i H } , solve the following. (a) Show that H = h x d i for some d that divides n . What is the order of H ? (b) Let a be any integer (not necessarily dividing n ). What is the order of x a ? (c) Conclude that all subgroups of G are cyclic, their order divides n , and more- over, for every divisor d of n , there is a unique subgroup of
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• Fall '17
• jane smith

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