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Fairy111
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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.
Fairy111 said:y^n.x^n.y^(-n) ?
Fairy111 said:that would be x^2
so (yxy^(-1))^n would be x^n
"ord(x)" represents the order of the element x in the group, which is the smallest positive integer n such that x^n = e, where e is the identity element of the group.
To "show" that ord(x) = ord(yxy^-1) in a group means to prove that the order of x and the order of yxy^-1 are equal. This involves demonstrating that both elements have the same smallest positive integer n that satisfies their respective equations, x^n = e and (yxy^-1)^n = e.
Proving that ord(x) = ord(yxy^-1) in a group is important because it helps us understand the structure and properties of the group. It also allows us to make conclusions about the relationship between the elements x and y and their powers, which can be useful in solving other mathematical problems.
One common method is to use the fact that for any element x in a group, ord(x) divides the order of the group. Another method is to use the properties of conjugacy, where ord(yxy^-1) is equal to ord(x). Other methods may involve using the group's identity and inverse properties to manipulate the equations and show that they are equivalent.
One example is the group of integers modulo n, where n is a positive integer. In this group, if x and y are relatively prime to n, then ord(x) = ord(yxy^-1) = phi(n), where phi(n) is the Euler phi function. This can be proved using properties of modular arithmetic and Euler's theorem.