Question

caun cu suugroup. (2) Let (a b) be a transposition in Sn, and let a E Sp. Prove that alaba-is another trans- position. Hint: Suppose that a(z) = a, a(y) = b for some x,y € {1,...,n). Now calculate ala b)a- applied to x, y, and z where z € {1,...,n} such that a(z) a or b. (9) def o lation

Answer 1

let oct Sn let 2(a)=n, a(b)=y forsone - myt BB {1,...,m} a la b) a (m) calab) a'd (a) [": n=2(a)] a alabas 2(6) I ca 5) la) =2) alab) a" (y) 'sala b) o d'a(6) [4=2(b)] = a(ab) (6) z dla) cm. . let Zt {1, 2, ..., h] sit. at my =) & '(Z) & 26x), of tabla (z) ally) aka : -) (3)¢ 4, 6. do : (ab)(a (2)) ='(7) [ a' (2) & a,b (a b) fines à (2)] alab) a '12) adal(2)=z.

a) Klabla! (z)= Z & ZE{1,..,h} 1. Sit. zzny a lab) & lon) ay . dlabl at ly)=m -). alaba? = (on y) which is a trams position