please send solution for part a only 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

The goal of this problem is to study the automorphisms of S, that is, the iso- morphisms from Ss to itself. Letf: Ss Ss be an automorphism. To lighten the notations, we denote by r the transposition (1 2) e S5. (a) Show that the permutation f(T) E S, has order 2.

The goal of this problem is to study the automorphisms of S, that is, the iso- morphisms from Ss to itself. Letf: Ss Ss be an automorphism. To lighten the notations, we denote by r the transposition (1 2) e S5. (a) Show that the permutation f(T) E S, has order 2.

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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Question

please send solution for part a only 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 lighten
the 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 two
disjoint 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 permutations
in S, that commute with 7 and the the number of permutations in S, that
commute 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,
namely
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) if
and only we are in one of the following situations:
o(a) = a and o(b) = b,
%3D
or
o(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, that
commute 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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