Question

Let wE S7 be a permutation which rearranges 7 objects as follows, showing the result on the lower line 2 3 4 6 7 5 5 4 2 7 6 1 3 a) Express was a product of disjoint cycles representing how each object moves Is w an even permutation, or an odd permutation? What is its order? products of disjoint cycles b) Calculate w3, w5 and w' 2 as c) Does there exist T E S7 for which T-lwr = (1234) (567) in cycle notation? If so, find two such permutations T; one the other an odd permutation. If not, give reasons should be an even permutation, d) Does there exist 7 E S7 for which r-1wT = (12345)(67) in cycle notation? If so, find two such permutations T; one should be an even permutation the other an odd permutation. If not, give reasons

Answer 1

2 3 4 56 5 1 2 7 (ow(1 5 6) (2 473) (1 5 6 s en even permmtatiom C2 4 73) am oold permutation am odd cncte n enen k eren yele is odd pererutation penmutation Rom ol2 5 6 ), 0 (2 4 7 3 = lom (,4 odol Ondar (b) w ( 6 62024 32TS 6)(2473) Ci562 02473 2.C2 3 74 1-4 0' (2 3 74 2 47 3)35 1 5 622 (1 56)5 (2 47 325 (156)(2 47 3) OC2 47 32=4 : (1 6 6)C2 47 3) CA524 3

(5 6 7) (1 2 3 - 7 52= J, C(6):5,(7) C (1) 2,C(2) 4,7(3)72(42 3 5)-wngne perrmudation permutation C C1 2 4 Eren (3) (1 2 3 4 6 7) jpermutation diffens in eyele as the d decombosition, amy agniraod,