Question

1. Consider the group U(35) = {1, 2, 3, 4, 6, 8, 9, 11, 12, 13, 16, 17, 18, 19, 22, 23, 24, 26, 27, 29, 31, 32, 33, 34} under the operation of multiplication modulo 35. The orders of the elements are: g 1112 341689|11|12|13|16|17|18|19|22|23|24|26|27|29|31 | 32 | 33 34 ord(g) | 1 | 12 | 12 62 463 12 4 3 12 12 6 4 12 6 6 4 2 6 12 12 2 a. Find two p-groups H and K such that U(35) is the internal direct product of H and K. Briefly explain why your groups work. b. Let H be the larger of the two groups above. Show how to decompose it as the internal direct product of (a) and H' where a is of maximal order and H' is some other subgroup of H. C. Using the decompositions above (perhaps repeating the second step as needed), write U(35) as the direct product of groups of the form Zpk (p prime).

Answer 1

1. 10(35) 1 = 24. (6) = { 1,63 (8) = { 1, 8, 29, 22}. * Then. U(35) = HK, were t = (6) *(8) = 2 x Z . = {(1,1),(1,8), (1,29), (1,22), (6, 1), (6,8), (6,29), (6,22) 1 . Clearly 141 = 8 = 23 and K = (11) = { 1, 11, 163 ~ Zz. 22 x Z hers Then, clearly Ank is drivial, since no subgroup of order 3. So my de composition work. D. Here, 4 = 467 x <87 ~ 22 x 22. 0 , 8)) x (4 (1) = cos x H', where a = (1,8), H' ={(6,1)) { H. This is work because, ca> nH'= {(1,1)]. 2020-4-6 11:52

. From @ 13) it follows that com U(35) ~ 2, * 24 * Z, = 22 x 2 x 23 This completes za