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Let G be a group of odd order. a) Prove that if x E G, then (x2) = (x). b) Showthatthemapp:G-Gsuchthatp(g)3g2 isabijection. c) If x and y are elements of G such that yxy-1 = x-1, show that x and y2 commute and that x = 1. %3D

Let G be a group of odd order. a) Prove that if x E G, then (x2) = (x). b) Showthatthemapp:G-Gsuchthatp(g)3g2 isabijection. c) If x and y are elements of G such that yxy-1 = x-1, show that x and y2 commute and that x = 1. %3D

Advanced Engineering Mathematics
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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Let G be a group of odd order. a) Prove that if x E G, then (x2) = (x). b) Showthatthemapp:G-Gsuchthatp(g)=g2 isabijection. c) If x and y are elements of G such that yxy-1 = x-1, show that x and y2 commute and that x 1.
Transcribed Image Text:Let G be a group of odd order. a) Prove that if x E G, then (x2) = (x). b) Showthatthemapp:G-Gsuchthatp(g)=g2 isabijection. c) If x and y are elements of G such that yxy-1 = x-1, show that x and y2 commute and that x 1.
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