Question

Let w e Sbe a permutation which rearranges 8 objects identified with letters, altering their positions to become as in the lower line of what follows: [A B C D E F G H (F DAEH C B G a) Express w as a product of disjoint cycles. Is w an even permutation, or an odd permutation? What is its order? b) Calculate wy, w and w-2 as products of disjoint cycles. c) Does there exist TE Sg for which 7-IWT = (1 2 345) (678) in cycle notation? If so, find at least two such permutations T. Can one be an even permutation, with another an odd permutation? If not, give reasons. d) Does there exist TE Sg for which r-IwT = (123456)(78) in cycle notation? If so, find at least two such permutations T. Can one be an even permutation, with another an odd permutation? If not, give reasons.

Answer 1

solution; Let wt Sp A B C D E F G H کے نا H C B G F D A G] a) was (A fc) FC) (BDE 4s (AF) (C) (80) (BC) (BH) (BG) written as produce transposetion. can Hence wu even even permutation is W Hence b) W] (A) (F) (G) CBHOG E) A CF) (C) (0) (E) (H) (G) ws - we (AFC) (BH D G €) w (63) (24 587) c) Since and Since (1 2 3 4 5) ( 648) Simidar permutation are 3

hence these exist I E Sg such that j'wt (1 2 3 4 5) (678) 2 j - (1 6 8) (2 3457) be not odd permutation TO can Similar because has permutation Cycle decomposition some d) There does not exists J E Sg So that J'wg (1 2 3 4 5 6 7 8 لا because LI and (1 2 3 4 5 6) (7.8) are not Similar permutation. o