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

[landor]

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...

 

[Gib Z]

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?

 

[PhaseShifter]

Is it necessarily true?
What about the case where x and y don't commute, but xⁿ=E for some specific n?

 

[Gib Z]

Sorry, I'm used to seeing n representing a general positive integer so I assumed that was the case here as well.

 

[PhaseShifter]

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.

 

[landor]

Thanks.