I do not understand the solution，please provide with more explanation，please analyze each step，especially the order and prime things FileEdit View History BookmarksProfilesTab Window Help今でもあなたはわたしの光K D D V O4)= c O L O Pi P c: 6 si P C:D E c. © trV Ja OMPEDG 2 M[E 目x EReatform.virdocs.com/r/s/0/doc/1559577/sp/165492544/mi/542334586?cfi=%2F4%2F2%5BP700101706000000000000000000ODBD%5D%2F20%5BP70..57+ -6.17 CorollaryIf a is a generator of a finite cyclic group G of order n, then the other generators of G are the elements of the form a', where r isrelatively prime to n.6.18 ExampleLet us find all subgroups of Z18 and give their subgroup diagram. All subgroups are cyclic. By Corollary 6.17 O, the elements 1, 5, 7, 11, 13, and 17are all generators of Z1s. Starting with 2,(2) = {0, 2, 4, 6, 8, 10, 12, 14, 16}.is of order 9 and has as generators elements of the form h2, where h is relatively prime to 9, namely, h = 1, 2, 4, 5, 7, and 8, so h2 = 2, 4, 8, 10, 14,and 16. The element 6 of (2) generates (0, 6, 12), and 12 also is a generator of this subgroup.We have thus far found all subgroups generated by 0, 1, 2, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 16, and 17. This leaves just 3, 9, and 15 to consider.(3) = {0,3, 6, 9, 12, 15},and 15 also generates this group of order 6, since 15 5-3, and the gcd of 5 and 6 is 1. Finally,(9)={0, 9}.The subgroup diagram for these subgroups of Z18 is given in Fig. 6.190.6.19 Figure Subgroup diagram for Z18.(1) =Z18m 1) Explana..pdfO Shafer Music of t..pdfO oliveros Sound.pdf3 Shafer Music of t.pdfa oliveros Sound.pdfShow AllAaMacBook ProG Search or type URL23&23478.WET.Y.

6.17 Corollary If a is a generator of a finite cyclic group G of order n, then the other generators of G are the elements of the form a", wherer is relatively prime to n. 6.18 Example Let us find all subgroups of Z1s and give their subgroup diagram. All subgroups are cyclic. By Corollary 6.17 O, the elements 1, 5, 7, 11, 13, and 17 are all generators of Z18. Starting with 2,. (2) = {0,2, 4, 6, 8, 10, 12, 14, 16}. is of order 9 and has as generators elements of the form h2, where h is relatively prime to 9, namely, h = 1, 2, 4, 5, 7, and 8, so h2 = 2, 4, 8, 10, 14, and 16. The element 6 of (2) generates (0, 6, 12), and 12 also is a generator of this subgroup. We have thus far found all subgroups generated by 0, 1, 2, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 16, and 17. This leaves just 3, 9, and 15 to consider. (3) = {0,3,6,9, 12, 15}, and 15 also generates this group of order 6, since 15 = 5-3, and the gcd of 5 and 6 is 1. Finally, (9) = {0,9}.

6.17 Corollary If a is a generator of a finite cyclic group G of order n, then the other generators of G are the elements of the form a", wherer is relatively prime to n. 6.18 Example Let us find all subgroups of Z1s and give their subgroup diagram. All subgroups are cyclic. By Corollary 6.17 O, the elements 1, 5, 7, 11, 13, and 17 are all generators of Z18. Starting with 2,. (2) = {0,2, 4, 6, 8, 10, 12, 14, 16}. is of order 9 and has as generators elements of the form h2, where h is relatively prime to 9, namely, h = 1, 2, 4, 5, 7, and 8, so h2 = 2, 4, 8, 10, 14, and 16. The element 6 of (2) generates (0, 6, 12), and 12 also is a generator of this subgroup. We have thus far found all subgroups generated by 0, 1, 2, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 16, and 17. This leaves just 3, 9, and 15 to consider. (3) = {0,3,6,9, 12, 15}, and 15 also generates this group of order 6, since 15 = 5-3, and the gcd of 5 and 6 is 1. Finally, (9) = {0,9}.

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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