Can x^2 Equal y in a Non-Abelian Group?

[gottfried]

Homework Statement

Let (G,.) be an non-abelian group. Choose distinct x and y such that xy≠yx.

Show that if x²≠1 then x²\notin{e,x,y,xy,yx}

The Attempt at a Solution

If x²=x would imply x.x.x^-1=x.x^-1 and x=e which cannot be.
If x²= xy or x²=yx would imply x=y which also cannot be.

How does one show that x²≠y?

 

[Dick]

Homework Statement

Let (G,.) be an non-abelian group. Choose distinct x and y such that xy≠yx.

Show that if x²≠1 then x²\notin{e,x,y,xy,yx}

The Attempt at a Solution

If x²=x would imply x.x.x^-1=x.x^-1 and x=e which cannot be.
If x²= xy or x²=yx would imply x=y which also cannot be.

How does one show that x²≠y?

x doesn't commute with y. Does x commute with x^2?

 

[gottfried]

Yes I believe it does so x².x=x.x² which would imply y.x=x.y if x²=y which is a contradiction.

Thanks for the help. You gave me just enough help to move forward with the problem but left some of the satisfaction of figuring it out to me which is cool, thanks.

 

[hedipaldi]

x^2=y→x^3=xy and x^3=yx