Question

(a) Let G be a cyclic group of order n. Prove that fo every divisor d of n there is a subgroup of G having order d. (b) Characterize all factor groups of Z70.

Answer 1

Uniqueness & Suppose His Subgroup of G such that another OCH'S = d Since t' is subgroup of Cyclic group G so h' will be generated by some power of a Let H'= <ar> by division algorithm po aq p=katr, oznak napt dp = dkat rd oedradk dkatda dka dr db a der dKq =) a a de ana = e a bd a dr 2 Now OCH') = ocab) = d Caba= e =) e from e = os dra dk But this implies dr=o oca=dk=n r=oas dzo and dren So p = kq So ha Cabz kas s Lak>=H But OCH') = OCH) H' cu so TH=h so uniquenese is proved and o Hence for din 7 unique subgsoup of order do

) G= Zro - it is generated by Lwith 2.1 = o ned 7e) Here the divisions of 70 75 (ع) (C cure 3670 ا ا ه ( 1257 Since a is چک all the group under addition Subgroups of G are of type < ka ? < .> = 2/> K Z'za for 35,76 ر۶/ رها,7 را 2 را = d ار2کارنا_الے ر 73 - so all factor groups are رو 202 و27 ك3 و والرد 22 ۱۹ و مهند 27رم 2 5 بهد77 Now 70 Zz = 702 کو Or وهو = م /2 35 به گ3 % ك3 رن کو : م / 42( 226ا 28 با عدد 2 ها وه و ره ک رد پا د 3 25 ره ره که 2 3 کروا 42 3 2 ادرلار محمو77 رہ کر والتوقيعر2هکلریلکي کوتS Z را دیدی؟ کہا ہم کو عمر 22 و 3 مقصد کرنی حياه لمحه27 وفيها 3 G Are all factor groups of G = 720