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Show that (Q*,-) is not cyclic..

Show that (Q*,-) is not cyclic..

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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I need help with #4.9
Section 4. Powers of an Element; Cyclic Groups 4.4 In (Z30.), find the orders of the elements 3, 4, 6, 7, and 18. 4.5 Let G be a group and let xEG be an element of order 18. Find the orders of X", 4.6 List all the elements of (Z45.) that are of order 15. 4.7 Let G-(x) be a cyclic group of order 24. List all the elements in G that are on order 4. 4.11 Is GL(2,R) cyclic? 41 4.8 The set of even integers forms a group under addition. Is this group cyclic? 4.9 Show that (Q*,-) is not cyclic. 4.10 Let G-(1,2,3,4,5,6) and define an operation O on G by aOb-ab, the remainder of ab (mod 7). For instance, 204-8-1, and 506-30-2. a) Show that (G,O) is a group. b) Is this group cyclic? 175 4.12 Consider the group (Z.), where a b=a+b-1. Is this group cyclic? 4.13 Show that if G is a finite group, then every element of G is ite group G such that every element of G has fin is of finite finite order
Transcribed Image Text:Section 4. Powers of an Element; Cyclic Groups 4.4 In (Z30.), find the orders of the elements 3, 4, 6, 7, and 18. 4.5 Let G be a group and let xEG be an element of order 18. Find the orders of X", 4.6 List all the elements of (Z45.) that are of order 15. 4.7 Let G-(x) be a cyclic group of order 24. List all the elements in G that are on order 4. 4.11 Is GL(2,R) cyclic? 41 4.8 The set of even integers forms a group under addition. Is this group cyclic? 4.9 Show that (Q*,-) is not cyclic. 4.10 Let G-(1,2,3,4,5,6) and define an operation O on G by aOb-ab, the remainder of ab (mod 7). For instance, 204-8-1, and 506-30-2. a) Show that (G,O) is a group. b) Is this group cyclic? 175 4.12 Consider the group (Z.), where a b=a+b-1. Is this group cyclic? 4.13 Show that if G is a finite group, then every element of G is ite group G such that every element of G has fin is of finite finite order
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