Question

(6)(20 points) (a) Let G be a cyclic group of order n. Prove that for every divisor dofn there is a subgroup of Ghaving order d. (b) Characterize all factor groups of Z70.

Answer 1

cyclic group of order 18 din Ill there if Subgroup of order d Sd- Now Cyclic G if n group of order then G zu dIn Now then of ostes d has element let and a EG oca) =d <as H= of is subgroep مه .. gb cyclic then din then has subgroup of corder d (6) Charactest cell factor group of Zio Sol" we know that factar group of Cyclic group Cyclic Zno is Cyclic and cell Subgroup zzo are Subgroup normal of of Zro Unique والے Zro & hos then Subgroup of order d 70 & d=1, 2, 5; 7 10, 14, 35,

let H, be 1 Subgroup of older subgroups of orgen 2 H2 be be H₃ Subgroup of order 5 T 7 / Hu 11 ܒ ܀ H5 14 71 35 17 11 70 7) H 7) 7 70 is Cydic g야 () Sle) - H 270 이 70 35 oG CH2) II () 와 " Co Z35 음 22 H 2 으 개 · 옵 (G) Ch 13 ~ Ziu 11 s 으 ( =) 유 G 얼리 HA - 7 ? ~ Z , ) 27 G Hy ( ) 이) ( ) 」 위의 70 ) 25 - 5 =) 0 HO 이 의 이음 7으 이 = 2 - G H Zz (( 35 N 2 0 0 는) 음