Question

1. Write the permutation o = (6, 1)(4, 2)(1, 2, 3)(5,8)(1, 2) of Sg as a product of independent cycles. Is o an element of Ag? Find the order of o. Find the inverse of o. Justify your answers.

Answer 1

= (6,1)(4.2) (1,2,3) (5.8) (1,2) To express 6 as a product at disjoint eycles, we have to keeping in mind that function composition is done from right to left and each cycle that does not contain a symbol fines the symbol We observe that, (1,2) sendo I to 2; (5,8) fines 2; (1,2,3) sendo 2 to 3; (4.2) fines 3; (61) fines b. so the net effect is o sends I to 3. This we begining with 3, 6=(13 .........Now repeating The entire process with 3 we have , cycle by eyele , right to left, 3 B B 1 1 6 , Then start with 6 6+6+6-76-76 → 1 i This we have a eyelé (1,3,6) Now start with 2 (Choose any except 1,3,6) 271 71 2 474 Then start with & 4 74 74 74 272 we have another eyele (2,4) Now start with 5 c choose any except 1,2,3,4, 575 -> 8+8 + 8 + 8 8 78 5 7 5 7 5 5 we have another eyele (5,8). And 77777777 pre (7) so we have, r= (1.3.6) (2,4) (5,8)(7) which is the product of disjoint eyelis lie, independent cycles

How, O= (1,3,6) (2,4), (5,8) (+) = (1,3,6) (2,4), (5,8) = (1,6), 11,6).(3,4) (5,8) ie 6 is product ab even number at transposition (a-2-excle called a transposition). so 6 is an even permutation Hone o is an element ab A8 CAS As is collection auf all even permutation), order at 8 = o(6) alcom (0(1,5,6), 0(2,4), (5,8),0 (7)) 1 = 1cm (3,2,2,1) E As erder aab r-cydinal e order ab 6 is 6.

The inverse permutation at o is 8= (F) (8,5)(4,2)(631) rby simply write each cyere ab o backword] Justification: To shows That S is inverce ab o we have to verify that oo=i, identity permutation Now, 68 = (15,6)(3,4) (5,8)(a) (7) (8,5) (4,2), (6,31) By similar procedure an in the previous starting from right to left we have (ebele by egcle) N6+66 67 66 61 272>4 4 4 4 4272 371711 1711 1B 4+4+2 2 2 2 2 4+4 5+5+5 8 8 8 5 55 6-73-3-3--33-333 6 777 77 77 77 8-78-78-75-75-35-8-88 on 00 = (136)(24) (58) (+)-(+)(8,5) (4, 2), (6,3,7) =i = (1)(2)(3) (4) (5)(6)(7)08) Henco 51-S (Justified)