Question

In Exercises 22 through 24, find all subgroups of the given group, and draw the subgroup diagram for the subgroups. 22. Z12 24. Z8 23. Z36

Answer 1

| Answer: 22) To find all subgroups of 2.2.1 Th2 = 20,1,2,3,4,5,6,7,8,9, 10, 11} 2112 is a cyclic group of order 12. hence all subgroups of 712 are also cyclic. By theorem of Lagrange, the order of a Subgroup of 12 divides 12. . 2112 has subgroups of orders 1, 2, 3, 4 6 & 12 Also we kown that for every divisor dy 12121 there exists a unique subgroup 4 g 442 g order d. a from all abore results, s o the following are the sabgroups of 212 . 1 2) 3) subgroup of order 1 Os 307 subgroup of order 2. <67= {0,5} subgroup of order 3. 347= {0,4,8} subgroup of order 4 <37= 20, 3, 6, 9 } a)

Date o subgroup y order 6 <27=20,2,4,6,8, 10} 6) subgroup of order 12 <17=212 eur * subgroup diagram na 212 <3)

23) To find all subgroups of Izs. The group Zizo have subgroups y orders 1, 2, 3, 4, 6, 9, 12, & 18. ago too - It's subgroup are. 1 g order 1 : <o>=10} y of order 2 - <187 = 20,18? 3 g order 3 i <127 = 10, 12, 24 } a) y order 4: <97= 20,9, 183, 278 of order 6: 567210, 6, 12, 18, 24,30% Toq order q:- <47=20,4,8,12, 16, 20, 24, 28,327 of order 12:- <37={0,3,6,9, 12, 15, 18, 21, 24, 27,30,33 8 g order 18:- <27=10,2,4,6,8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 30 32, 34) 19 q order 36:- <17 = Igr.

subgroup diagram of 236

for, (24) To find all subgroups of Z8 . - The group 418 have the subgroups of orders , 1, 2, 4, 6 8 Its subgroups are. D of order 1 - <0>= 10}. is of order 2 :- <47=20,4} 3) g order hir (27=20, 2, 4, 6} 2 ;- a) of order 8 :- <17=418. * subgroup diagram.

Answer 2

If N normal G prove order (aN)/order (a)€a belong G