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.
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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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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 & G by
fa,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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff6d69502-ffbd-48d7-85dd-a2e3610b14b4%2F31bd1e2e-510f-405c-a7d8-2b8d511e465e%2Ffvgm686_processed.png&w=3840&q=75)
Transcribed Image Text: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 & G by
fa,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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