in those partice question could you help me to fix my mistake

here is my work

since Ais has element of ( mhumn ) order by counter example Lom ( 7. 5. 1 )= 35 2 ? so Ais has no element of order 42 do & a = 0 .... OK , disjoint gale 10 / = LCM ( 10. 1 . ... 18 x 1 ) = 42 - 10:1 / 42 2 2 10:1 = $ 1. 2, 3, 6. 75 10:1 = 13 ) . . .Take 5= ( 12) 13.) 2 = ( 2 , 3 ) 14.5 ). let 6 , Q G H => 6 = 02 = E Also 6 , a both are disjoint X Then o (6 2 ) = lom Colby , . (as ) 2 dom ( 2, 2), 6* E, d+E 22 62 GAR & Azis finite. and God GH B group. So Hmust be subgrey & At and a close property9:004 ..1 4G A 2021 F 10 A 248.pdf . . . 1) 6 = ( 1 , 12 , 12 , 6 , 2 , 5 ) ( 3,6, 10, 14) ( 4, 15, 9, 13, 18, 11, 17, 7, 20, 165 161 = lem ( 6, 4, 10) = 60 161= 60 2) since 161260 is even number so the permeteen is odd. NO. 6 n odd 1 (', 2) 13, 4) 1 -2 I ever permutation Even Cycle 2 01 permutation 5 2 3 + old permutation 4x, FY 14 2 old 8 101 R even FX\f9:004 ..1 4G A 2021 F 10 A 248.pdf . . . x = ( 2 8 4 9 25 ) ( 1 10 6 3 ) at ) = Icm ( 6, 4) = 12 -> KOV / = 12 B = ( 247 ) ( 5 6 9 ) ( 13610) O( B ) = Icm ( 3, 3, 4 ) D ( 13 ) = 12 =7 (432 1 = 12 = 2 LB = ( 284 9 75 ) (1 10 63 > ( 2 4> > (389) ( 136102 2 3 5 4 8 7 8 9 10 = ( 1 9 3 4 5 6 8 7 2 10) = ( 29 ) 145) 1 2 8 ) ocLB ) = lom co ( 29), 0145, 0(781 ) - lcm ( 2, 2, 2 ) - 2 by asing OCHR ) = DCH JO (k) . J => 2 - 0( ) . 0 ( > 0 ( n > = ) 1 ) = 12 4 1 2 12=72 12 271 = 12 s Ld> 2 14 9 KE 72 ? * Class 9 92 52 ) G be group a, bGG 1a1= 20 ( 61 = 18 & Cann + Le3 let x G La> n X = qR x=b , xte 02k-20, 05 k - 18 since order of a ze so order of al can be 2, 5, 10, 20 =7 1X1 = (2), (8), (10), or ( 20) order of 6* can be 2 , 3 , 6 , 9 , 18 (X1 = ( 27, laR1 = 2 20 ged ( 20, 12 ) = 2 god ( 2, 12) = 102 0E k - 20 R = 10 1x1 =2 18 #1 = 2 16 good ( 18, ) = 2 good ( 1 8 2 ) = 9 7 05 k - 8 R = 9 Since X= a = 10 X= b' = by X = = ble x = a = b9\f9:004 ..1 4G A 2021 F 10 A 248.pdf . . . 6 = ( 3, 2 , 5,6 ) ( 64) = 1 ( 6) = 12 why ? explain 1641- 101 grallol , 4) 161 = 160 2) ... since Sis has no element of order 16 so there does not east an element 6 E Sis, Sit 6 = ( 3, 2, 5, B)[1) Does there exist an element of order 42 in A13? Justify your answer [2) Let G be a group, (1,5 E G and |n| = 20, M = 18 and {a} F] {b} 79 {e}. Prove that am : 59. [3) Let H C (Q, +} be a subgroup generated by nitely many elements. Prove that H is cyclic. [4) Does there exist an element [If E 315 such that J4 = (3 2 5 6)? Justify your answer. (5) Let 1234567891011121314151617181920 1256151102021314171918394711816 [3.) Find |cr|; (b) Is 0' even or odd? [5) Let 0: = (1 24){352 1)(542) Find 0:99. (7) Let or = {284975){11063} and 5 = [247](589)(13610) be ele ments in Sm. Find Her) F1 ()|. Justify your answer. [8) Let H = {or E A7 | 02 = (-3}. Is H a subgroup of A7? a