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10. (a) If gh hg in a group and o(g) and o(h) are finite, show that o(gh) is finite.

10. (a) If gh hg in a group and o(g) and o(h) are finite, show that o(gh) is finite.

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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### Transcription **Problem 10. (a):** If \( gh = hg \) in a group and \( o(g) \) and \( o(h) \) are finite, show that \( o(gh) \) is finite. *From an abstract algebra class* --- ### Explanation This problem is from an abstract algebra context. It addresses properties in group theory, specifically related to the concept of element orders in a group. The notation \( o(g) \) represents the order of the element \( g \), which is the smallest positive integer \( n \) such that \( g^n = e \), where \( e \) is the identity element. The problem asks for a proof that the order of the product \( gh \), which commutes (\( gh = hg \)), is finite if the orders of \( g \) and \( h \) are finite. This leverages the concept of element commutativity and their finite orders to deduce the form of the solution.
Transcribed Image Text:### Transcription **Problem 10. (a):** If \( gh = hg \) in a group and \( o(g) \) and \( o(h) \) are finite, show that \( o(gh) \) is finite. *From an abstract algebra class* --- ### Explanation This problem is from an abstract algebra context. It addresses properties in group theory, specifically related to the concept of element orders in a group. The notation \( o(g) \) represents the order of the element \( g \), which is the smallest positive integer \( n \) such that \( g^n = e \), where \( e \) is the identity element. The problem asks for a proof that the order of the product \( gh \), which commutes (\( gh = hg \)), is finite if the orders of \( g \) and \( h \) are finite. This leverages the concept of element commutativity and their finite orders to deduce the form of the solution.
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