A particular subgroup of a Free Group is normal

[empyreandance]

Hello friends,

I'm working through my book and I'm having a lot of trouble coming to terms / believing this. Could anyone assist?

Let F be a free group and N be the subgroup generated by the set {x^n : x is in F and n is fixed} then N is normal in F.

Any ideas?

 

[micromass]

Use this trick:

$$yx^ny^{-1}=(yxy^{-1})^n$$

 

[empyreandance]

You saved me yet again! Thank you friend.