Question

4. List all left cosets of An in Sn. (See 3.7.11.) For a given permutation o in Sn, how can you tell from o which coset o An is? Example 3.7.11. Pick a positive integer n > 2 and consider the group S. We define An = {o ESO is an even permutation). We will use the first theorem above to verify that An is a subgroup of S First of all, the identity is defined to be an even permutation, so it is in H. Now suppose that o andare tuo even permutations. We noted back in 3.3.16 that their product is even. So we have on is in H. Finally, suppose that o is in H. We must show that o-l is also in H. As o is in H, it is even and so we may write = 0102...0k where each o; is a transposition and k is even. Every transposition is its own inverse and, using 3.1.17, we have o = (01...08) = 0...01 and we have written o as a product of an even number of transpositions and so it is in An as desired. We have verified the three conditions of the first theorem above and so A. is a subgroup of S.

Answer 1

la In l Let In where us us consider group Consider the . the an permulations on n Zeller and Anso csn la is an even permutation } . we know that Ans Sn. let now us consider the set we rechthon {-1,1} define on the set {-1,17 by and 16-1)-(-0.1 = -1 1.1= -1).(-1) - 1 ano then the det f-1,13 bener becomes a group under this, on with element 1. identity now . if a E In then there esist trum.position. Tur. T K E Sn such that a = 1, T2-.. Tk now if there exist another set of transpositions, titz , txe sh such that a = titz-..tk ther k=k' (med 2).. now let us consider the function sgn: Sn 4-11pp a gn(a) = 60 whenever w=6772. RK cohere 7172, ., TK are transpositions now a Ps an even permutation if and only if, Ign (6) = 1 and a is an odd permutation if and only if Sgn (0) = -1 claim. If allaresh then agn (or az) = dgn(a, Dign (12) Proof. - Suppose aisarean. then there exist sets of transpositions allti2... Tir and Tu T2 Tzs such that - ai = 211 212 - Tir and a2 = Tu Tu Tu then agar = T1 T2 Tr Tu Tu as then slyn (0) = -1)", dgn (02) = 6198 and dgn (6Y0/2) = 6*** = 60 61)' = sgo (wy) egn(n) cs ansign Cajaz) = dgncar) agacoes when ever algas Sn-

defined as a let us consider the function O: Sn (£1,13, o) o(a) = lgn (a) wheneuer a esni let onare Sn then olmaz) = dgn (ojai) = ogn (as) egn(Qc) = 0(0) o(a) 0(0,0) = 0 (0) 0 (02) whenever apore Sn s o is a homomorphism now Ocid) = ogn (id) = sqr((1 2)(1 2)) =60°= 1 Lince id = (12/12). salget fiz. and (1 2))= dgn((1 2)) = 60's-1 This implies o is an ep epimorphism Conto). now Ker(0) - {ae Son loca) = 1 } 1 = { a Esn I agn (0) = 1} = An → Kerco) - An then by the first theorem of isomorphism we have - Sn / Ana ({-1, 1},.) → Isni Anl - 2 → Sn/An = {. An , (I 1) Anį now suppose a En now either agn (0) = 1 or dyn (a) = -1 if dgn (@) = 1 → at An → aan = An if dgn (a) = -1 a is an odd permutation so there exist odd numbers of Titr. transpositions 2x+1 Such that a = TITL Tekti then (12)'a = (23907 22Tilkt = (21) 77 Te-. Texte dince dgn (62)'a) = (-1)24+2 = 1 = a (2) la E An → aAn = (12) An so if sgn (0) = 1 then a An = An CS Scaifeddgn(~) = -1 then aan = () An CamScanner