How to prove that if x^n*y=y*x^n, then x and y commute

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In summary, the conversation discusses the concept of commuting elements in a group, where x and y commute if x*y = y*x. The question then arises of how to make the equation y*x^n = x^n*y represent the same concept. It is noted that this is not necessarily true if x and y do not commute, but it can be true if xn = E for a specific value of n. The confusion around the use of n as a general integer is also addressed.
  • #1
landor
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Let x and y be elements of a group.

I can see that it works the other way, i.e. if x and y commute, then y*x^n = x^n*y...
 
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  • #2
Write down the equation that represents x and y commuting, and then the equation you have beneath it. How can you make the second one become the first?
 
  • #3
Is it necessarily true?
What about the case where x and y don't commute, but xn=E for some specific n?
 
  • #4
Sorry, I'm used to seeing n representing a general positive integer so I assumed that was the case here as well.
 
  • #5
Well, the problem with that is n either has a specific value or is a general integer...if it's a general integer, then it's trivial to consider the case n=1. So I'm assuming that's not what is meant.
 
  • #6
Thanks.
 

1. How do I prove that x and y commute if x^n*y=y*x^n?

To prove that x and y commute, we need to show that xy = yx. We can do this by using the given equation x^n*y = y*x^n and manipulating it algebraically.

2. What is the significance of x^n*y=y*x^n in proving commutativity?

The equation x^n*y=y*x^n is significant because it shows that the order in which x and y are multiplied does not matter. This is a key property of commutativity.

3. Can I use the commutative property to prove x^n*y=y*x^n?

No, you cannot use the commutative property to prove x^n*y=y*x^n. The commutative property only applies to addition and multiplication, not exponentiation.

4. Are there any other properties or theorems that can be used to prove x^n*y=y*x^n?

Yes, there are other properties and theorems that can be used to prove x^n*y=y*x^n. One example is the associativity property, which states that (xy)z = x(yz). This can be used to manipulate the given equation and show that xy = yx.

5. How can I apply this proof in real-world situations?

The proof of x^n*y=y*x^n can be applied in various situations, such as in algebraic equations, mathematical proofs, and computer programming. It can also be used to simplify and solve problems involving exponents and variables.

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