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

[Fairy111]

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.

 

[Dick]

What's (yxy^(-1))^n?

 

[Fairy111]

y^n.x^n.y^(-n) ?

 

[Dick]

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)).

 

[Fairy111]

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

 

[HallsofIvy]

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.

 

[Dick]

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!