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6. Consider the set G = {4, 8, 12, 16}. Show that this set is a group under multiplication modulo 20 (construct its Cayley table). What is the identity element? Is this group cyclic? If so, find all of its generators.

6. Consider the set G = {4, 8, 12, 16}. Show that this set is a group under multiplication modulo 20 (construct its Cayley table). What is the identity element? Is this group cyclic? If so, find all of its generators.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter3: Groups
Section3.2: Properties Of Group Elements
Problem 10E: 10. Prove that in Theorem , the solutions to the equations and are actually unique. Theorem 3.5:...
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1:31 e s3.amazonaws.com Chapter 4 Homework 1. Find all the generators of 2» 2. List the elements of the subgroups H = <3> and K = <5> in . 3. Let a be an element of a group G with |a| = 15. Find the order of the following elements of G: - |. |). \| 4. Let G = <a> and |a| = 24. List all generators for the subgroup of order 8. 5. Determine the subgroup lattice for Z 6. Consider the set G = {4, 8, 12, 16}. Show that this set is a group under multiplication modulo 20 (construct its Cayley table). What is the identity element? Is this group cyclic? If so, find all of its generators. 7. Prove that a group of order 3 must be cyclic. 8. For any element a in a group G, prove that <a> is a subgroup of C(a). 9. Let G be a cyclic group with |G| = 24 and let a be an element of G. If a' - elidentity) and a" - e(identity) show that G = <a>. 10. Show that the group of positive rational numbers under multiplication is not cyclic. H 11. Prove that ""ilo 1f=<} is a cyclic subgroup of GL(2, R)
Transcribed Image Text:1:31 e s3.amazonaws.com Chapter 4 Homework 1. Find all the generators of 2» 2. List the elements of the subgroups H = <3> and K = <5> in . 3. Let a be an element of a group G with |a| = 15. Find the order of the following elements of G: - |. |). \| 4. Let G = <a> and |a| = 24. List all generators for the subgroup of order 8. 5. Determine the subgroup lattice for Z 6. Consider the set G = {4, 8, 12, 16}. Show that this set is a group under multiplication modulo 20 (construct its Cayley table). What is the identity element? Is this group cyclic? If so, find all of its generators. 7. Prove that a group of order 3 must be cyclic. 8. For any element a in a group G, prove that <a> is a subgroup of C(a). 9. Let G be a cyclic group with |G| = 24 and let a be an element of G. If a' - elidentity) and a" - e(identity) show that G = <a>. 10. Show that the group of positive rational numbers under multiplication is not cyclic. H 11. Prove that ""ilo 1f=<} is a cyclic subgroup of GL(2, R)
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