Homework Help: Proof help

1. Nov 5, 2006

raj123

If G is a group and x, y are in G. Show that o(x) = o(y^-1xy), where o(x) means order of x.

thanks

Last edited: Nov 5, 2006
2. Nov 5, 2006

3. Nov 5, 2006

0rthodontist

Why not try computing a few powers of y-1xy and see if that gives you any clues?

4. Nov 6, 2006

raj123

isn't y-1xy=x since y-1y=e. so o(y-1xy)=o(x).

5. Nov 6, 2006

0rthodontist

Why would you think that? Not all groups are commutative.

What is (y-1xy)2 equal to?

6. Nov 6, 2006

matt grime

Do not assume your group is commutative unless expressly told it is. yxy^-1 is most definitely not the same as x in general, and rule number 1 is that you should never assume otherwise unless told you may. I'm sure this was hammered home in your first lesson (and if not it should have been).

Now, please try to follow the hint given. What is (yxy^-1)^2? (yxy^-1)^n?

Remember that the order of z is the least positive n with z^n=1, so it obviously suffices to show that the order of z is less than the order of x, since by symmetry the order of x is then less than the order of z, so they are equal.

You should go through that argument a couple of times to see why it is true, as well.

7. Nov 6, 2006

raj123

(y-1xy)2 =y-2x2y2 =y-2+2x2=x2

8. Nov 6, 2006

0rthodontist

I don't think you're getting it. Could you provide what you think the justification is for each of those steps? This means working directly from the group axioms. You have three equal signs, which means you'll need three justifications.

There's more than one problem here. First you say that (y-1xy)2 =y-2x2y2. That isn't necessarily true. Then you say that y-2x2y2 =y-2+2x2, which again is not necessarily true.

To get you started,
(y-1xy)2 = (y-1xy)(y-1xy). What does that equal? You can use the associative law and the definition of inverses.

Last edited: Nov 6, 2006
9. Nov 6, 2006

matt grime

For the fourth (?) time: groups are not to be presumed commutative, you cannot do this. xy=/=yx. You cannot just swap round the order of x and y at will.

10. Nov 6, 2006

raj123

(y-1xy)(y-1xy) = y-1xyy-1xy = y-1x2y

Last edited: Nov 6, 2006
11. Nov 6, 2006

matt grime

Yes. Now what about (yxy^-1) raised to the n'th power?

12. Nov 6, 2006

raj123

that will be...
(y-1xy)n = y-1xny

13. Nov 6, 2006

NateTG

And, now what happens when $n=o(x)$?

14. Nov 6, 2006

mathwonk

1) conjugation is an isomorphism.
2) isomorphisms preserve order of elements.

15. Nov 7, 2006

raj123

xn= e.what are the next steps in this proof?

16. Nov 7, 2006

matt grime

Can you please try and put two and two together? You're asked to raise something to the power n, and then asked to consider what happens when n is the order of x. Now, please, try to think what that might mean.

17. Nov 7, 2006

raj123

thanks. that was easy.

18. Nov 7, 2006

0rthodontist

You're not done yet, though, because you also must show that no smaller positive power of y-1xy equals e. Pick the smallest power (y-1xy)k that equals e, and show that k = |x|. (the order of x)

19. Nov 8, 2006

HallsofIvy

You just said "xn= e". Use that in your formula (y-1xy)n= y-1xny!

However, o(x) is defined as the smallest n such that xn= e. Just showing that (y-1xy)n= e is not enough to say that the order of y-1xy is n.