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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.
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Why not try computing a few powers of y^{-1}xy and see if that gives you any clues?raj123 said:If G is a group and x, y are in G. Show that o(x) = o(y^-1xy), where o(a) means order of a.
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0rthodontist said:Why not try computing a few powers of y^{-1}xy and see if that gives you any clues?
Why would you think that? Not all groups are commutative.raj123 said:isn't y^{-1}xy=x since y^{-1}y=e. so o(y^{-1}xy)=o(x).
(y^{-1}xy)^{2} =y^{-2}x^{2}y^{2} =y^{-2+2}x^{2}=x^{2}0rthodontist said:Why would you think that? Not all groups are commutative.
What is (y^{-1}xy)^{2} equal to?
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.raj123 said:(y^{-1}xy)^{2} =y^{-2}x^{2}y^{2} =y^{-2+2}x^{2}=x^{2}
raj123 said:(y^{-1}xy)^{2} =y^{-2}x^{2}y^{2} =y^{-2+2}x^{2}=x^{2}
NateTG said:And, now what happens when [itex]n=o(x)[/itex]?
matt grime said: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.
The order of x refers to the smallest positive integer n such that x^n = e, where e is the identity element. Essentially, it is the number of times you have to multiply x by itself to get back to the identity element.
To prove that o(x) = o(y^-1xy), we must show that the two expressions have the same value. This can be done by showing that x^n = e if and only if (y^-1xy)^n = e. This can be done using algebraic manipulations and the properties of group operations.
Sure, let's say we have a group G with elements x and y. If o(x) = 4, this means that x^4 = e, where e is the identity element. Now, if we take y^-1xy, this is essentially the same as saying that we are replacing x with y in the expression x^4. If o(y^-1xy) is also equal to 4, then this means that (y^-1xy)^4 = e. This shows that o(x) = o(y^-1xy) in this specific example.
The concept of conjugacy in group theory is closely related to this proof. Conjugacy essentially means that two elements are related to each other by a change of basis or perspective. In this case, we are showing that x and y^-1xy are conjugate to each other, which is why they have the same order.
Yes, this proof is applicable to all groups. The concept of order is a fundamental property of groups and the proof relies on the basic principles of group operations. Therefore, it can be applied to any group, regardless of its specific structure or elements.