(a) Consider the symmetric group Sx on the set X = R-{0, 1}. Let f be the function defined by the following rule I - 1 f(x) = %3D for all r E X. Show that f E Sx, and find the order of f and the inverse of f. (b) Let G be a group of order 72. Show that if G is not cyclic, then for any g EG either g24 = 1 or g36 = 1. (c) Let G be a group of order 60 and g be an element of G of order 10. Show that there is no element r of G satisfying r = g.
(a) Consider the symmetric group Sx on the set X = R-{0, 1}. Let f be the function defined by the following rule I - 1 f(x) = %3D for all r E X. Show that f E Sx, and find the order of f and the inverse of f. (b) Let G be a group of order 72. Show that if G is not cyclic, then for any g EG either g24 = 1 or g36 = 1. (c) Let G be a group of order 60 and g be an element of G of order 10. Show that there is no element r of G satisfying r = g.
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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Transcribed Image Text:(a) Consider the symmetric group Sx on the set X = R-{0, 1}. Let f be the function defined by the following
rule
x - 1
f(r) =
%3D
for all r E X. Show that f E Sx, and find the order of f and the inverse of f.
(b) Let G be a group of order 72. Show that if G is not cyclic, then for any g EG either g24 = 1 or g36 = 1.
(c) Let G be a group of order 60 and g be an element of G of order 10. Show that there is no element r of
G satisfying r = g.
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