please send solution for part fonly handwritten solution accepted Problem 3. The goal of this problem is to study the automorphisms of S, that is, the iso-morphisms from S5 to itself. Letf: SsSs be an automorphism. To lightenthe notations, we denote by r the transposition (1 2) e S.(a) Show that the permutation f(r) 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 r 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 a E Ss, the permutation o o (a b)00 is also a transposition,namelyo (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,%3Doro(a) = b and o(b) == a.How many permutations in Ss 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

Given: part( f ): To prove fτ is a transposition.

Answered: Use the previous parts to deduce that…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.5: Isomorphisms

Problem 20E: For each a in the group G, define a mapping ta:GG by ta(x)=axa1. Prove that ta is an automorphism of...

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In Example 3, the group S(A) is nonabelian where A={ 1,2,3 }. Exhibit a set A such that S(A) is abelian. Example 3. We shall take A={ 1,2,3 } and obtain an explicit example of S(A). In order to define an element f of S(A), we need to specify f(1), f(2), and f(3). There are three possible choices for f(1). Since f is to be bijective, there are two choices for f(2) after f(1) has been designated, and then only once choice for f(3). Hence there are 3!=321 different mappings f in S(A).
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Elements Of Modern Algebra

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Elements Of Modern Algebra

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9781285463230

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Use the previous parts to deduce that f(7) is a transposition.