Exercise 3. Let T be a set such that |T| ≥ 3 and let G = Sym(T) be the group ofpermutations of T.(i) For each pair a b € T, define fa,b & G byfa,b (t)=b,a,if t = a;if t = b;t,Prove that C(fa,b) = { g = G|{g(a), g(b)} = {a,b} }.(ii) Prove that Z(G) = {I}.otherwise.

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Answered: Exercise 3. Let T be a set such that…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Exercise 3. Let T be a set such that |T| ≥ 3 and let G = Sym(T) be the group of permutations of T. (i) For each pair a b € T, define fa,b E G by b, fa,b(t) = a, t, G|{g(a), g(b)} = {a,b}}. if t = a; if t = b; Prove that C(fa,b) = {g (ii) Prove that Z(G) = {I}. otherwise.