In a group, show that ord(x) = ord(yxy^-1)

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Homework Help Overview

The problem involves demonstrating that the order of an element x in a group G is equal to the order of the conjugate element yxy^-1. The discussion centers around group theory concepts, particularly the properties of group elements and their orders.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants explore the expression (yxy^-1)^n and its implications for the order of x. Questions arise regarding the simplification of this expression and its relation to the definition of order in group theory.

Discussion Status

The discussion is active, with participants providing hints and corrections regarding the manipulation of the expression. Some guidance has been offered about simplifying the expression, but there is no explicit consensus on the next steps or the relationship to the definition of order.

Contextual Notes

Participants are navigating through the implications of group properties and the definition of order, with some expressing uncertainty about the correct simplifications and assumptions involved in the problem.

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Homework Statement



If x and y are elements of a group G show that ord(x)=ord(yxy^-1)

Homework Equations





The Attempt at a Solution



Some hints to how to do this would be great.
 
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What's (yxy^(-1))^n?
 


y^n.x^n.y^(-n) ?
 


Fairy111 said:
y^n.x^n.y^(-n) ?

You aren't going to get anywhere taking wild guesses. Try simplifying (yxy^(-1))^2=(yxy^(-1))(yxy^(-1)).
 
Last edited:


that would be x^2
so (yxy^(-1))^n would be x^n
 


Yes, that is correct. Now, what does that have to do with the definition of "order"?

Oops! Dick is correct. I forgot about the first and last y and y-1!

But, "what does that have to do with the definition of 'order'" still stands.
 
Last edited by a moderator:


Fairy111 said:
that would be x^2
so (yxy^(-1))^n would be x^n

No. (yxy^(-1))^n would be yx^ny^(-1), NOT x^n. Be careful!
 

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