1. Let T and (abc) be permutations in a symmetric group S. Prove thatτ (abc)τ1 = (τ(α) + τ(b) + τ(c)My wrong answerLet T E Sn be any permutation(a, b, c) E S be any 3-cycleProve: T(a b c)r-1 = (t(a) + T(b) r(c))(a, b, c) E Sn meansb, bc, c aSo to showа —aT(a b c)r is the 3 - cycle:τ (α) τ(b), τ ( b) > τ(c), τ(c) > τ(α)CBy changing the labels a, b and c we may assume that the 3-cycle is (123). Now anypermutation t is a product of transpositions, so it is enough to prove the statementwhen T is a transpositon --that is ,T changes only two indices, leaving the others fixedT.Case 1: T does not involve any of the indices 1,2 or 3. In this case T commutes with (123)and there is nothingr =(45)). Note that (45) (45) identityto prove. (we may assume TpermutationT(1 2 3)r= (45)(1 2 3)(45) = (45)(45)(1 2 3)= (1 2 3) (T(1)r(2)t(3))As t fixes (1,2, 3)-1Case 2: assumeT is a transposition involving 2 of the symbols 1, 2,3. say T (12)T(1 2 3)r= (12)(1 2 3) (12) (132) (213)(T(1)r (2)7(3))-1=Case 3: T involves onlyone of the symbols 1,2 or 3. Assume t -(14). That is, r takes 1 to4 and 4 to 1T(1 2 3)r(14) 1 2 3) (14) = (234) = (423)- (τ(1)τ (2)τ (3))= as t fixes 2,3 and t(1) fixes 4

Question

I need help understanding attached question for abstract algebra. My answer teacher said dosnt accountt for all numbers that may be in a set, i only used up to 5. Please help and show me correct way

Step 1

Given two permutations in a symmetric group are

Step 2

Consider, the permutation of 3-cycle as

Step 3

Now take j such that j belo...

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