﻿ Problem Set 3: Problem 4
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เกียรติศักดิ์
neutrino

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 « on: November 16, 2006, 11:22:12 PM »

Let be an element of a finite group . Show that .

The author seems to forget to tell that must also be a finite group whose order is a prime number, since all element don't necessarily have the same order, unless it has no subgroup. Am I correct?
 « Last Edit: November 17, 2006, 12:10:12 AM by เกียรติศักดิ์ » Logged

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SuperHelper

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 « Reply #1 on: November 17, 2006, 07:22:38 AM »

...
The author seems to forget to tell that must also be a finite group whose order is a prime number, since all element don't necessarily have the same order, unless it has no subgroup. Am I correct?

No, the problem is complete as it is.

Hint: For any element of a finite group, consider its period. Show that the perioad must form a subgroup. Then use Lagrange's theorem.
 « Last Edit: November 17, 2006, 12:05:34 PM by ปิยพงษ์ - Head Admin » Logged

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เกียรติศักดิ์
neutrino

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 « Reply #2 on: November 17, 2006, 10:10:39 AM »

Oh I see!

(I overlooked the fact that the order of an element divides the order of a group, i.e. , where is an integer and is the order of the element. Hehe. )

Thank you.
 « Last Edit: November 17, 2006, 10:13:18 AM by เกียรติศักดิ์ » Logged

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