prob 3 part fplz provide handwritten solution for part B asap Problem 3. The goal of this problem is to study the automorphisms of S, that is, the iso-morphisms from S5 to itself. Let f: S5 Ss be an automorphism. To lightenthe notations, we denote by r the transposition (1 2) € S,.(a) Show that the permutation f (7) E S, has order 2.(b) Use part (a) to show that f(r) is either a transposition or a product of twodisjoint transpositions.(c) Show that a permutation o E S, commutes with 7 if and only if the permu-tation f(o) commutes with f(7). Deduce that the number of permutationsin S, that commute with 7 and the the number of permutations in S, thatcommute with f(r) are equal.Let (a b) e S, be a transposition, with 1 < a, b < 5 distinct. We recall that,for a permutation o e Ss, the permutation o o (a b) o01 is also a transposition,namelyo o (a b) o o- = (0(a) o(b))(This was proved in Problem Sheet 7, you do not have to prove it here.)(d) Show that a permutation o e S; commutes with the transposition (a b) ifand only we are in one of the following situations:o(a) = a and o(b) = b,%3D%3Dora(a) = b and o (b) = a.%3DHow many permutations in S; commute with the transposition (a b)?(e) Use a similar strategy to determine the number of permutations in S, thatcommute with a product of two disjoint transpositions of the form (a b)(c d)with 1< a, b, c, d < 5 all distinct.(f) Use the previous parts to deduce that f(r) is a transposition.

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Answered: (b) Use part (a) to show that f (T) is…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.3: Subgroups

Problem 26E: Let A be a given nonempty set. As noted in Example 2 of section 3.1, S(A) is a group with respect to...

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(b) Use part (a) to show that f (T) is either a transposition disjoint transpositions. or a product of two