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Let G be a group and let a, b E G. (a) Prove that o(ab) = o(ba). (Note that we are not assuming that a and b commute. In fact, the interesting case is the case where a and b do not compute.)) (b) Prove that o(a-lba) = o(b). %3D (c) If ab = ba and gcd(o(a), o(b)) = 1, then prove that o(ab) = o(a)o(b).

Let G be a group and let a, b E G. (a) Prove that o(ab) = o(ba). (Note that we are not assuming that a and b commute. In fact, the interesting case is the case where a and b do not compute.)) (b) Prove that o(a-lba) = o(b). %3D (c) If ab = ba and gcd(o(a), o(b)) = 1, then prove that o(ab) = o(a)o(b).

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 and let \( a, b \in G \). (a) Prove that \( o(ab) = o(ba) \). (Note that we are not assuming that \( a \) and \( b \) commute. In fact, the interesting case is the case where \( a \) and \( b \) do not commute.) (b) Prove that \( o(a^{-1}ba) = o(b) \). (c) If \( ab = ba \) and \(\gcd(o(a), o(b)) = 1\), then prove that \( o(ab) = o(a)o(b) \).
Transcribed Image Text:Let \( G \) be a group and let \( a, b \in G \). (a) Prove that \( o(ab) = o(ba) \). (Note that we are not assuming that \( a \) and \( b \) commute. In fact, the interesting case is the case where \( a \) and \( b \) do not commute.) (b) Prove that \( o(a^{-1}ba) = o(b) \). (c) If \( ab = ba \) and \(\gcd(o(a), o(b)) = 1\), then prove that \( o(ab) = o(a)o(b) \).
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