Show that (Q*,-) is not cyclic..
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F27b38c32-736b-4354-8263-82b5dd937d13%2F40230e2a-7c96-41dd-ae83-269e38a23f81%2Fd6y0jsa_processed.jpeg&w=3840&q=75)
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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