Originally Posted by

**simplependulum** Hi I am new to Group Theory and I am confused by a problem :

Let $\displaystyle G $ be any group of order $\displaystyle n $ and $\displaystyle k $ be an integer prime to $\displaystyle n $ .

Is it true that $\displaystyle x^k = y^k ~ \implies x = y ~~~ x,y \in G $ ?

I know there exists $\displaystyle A,B ~\in \mathbb{N} $ such that $\displaystyle Ak - Bn = 1 $

Then $\displaystyle (x^k)^A = (y^k)^A $

$\displaystyle x^{Ak} = y^{Ak} $

$\displaystyle x^{1 + Bn} = y^{1 + Bn} $

$\displaystyle x (x^n)^B = y (y^n)^B $

I think if $\displaystyle a^n = e $ , then i can prove it but is it really true ? Or my proof has some mistakes ? Actually , it is what i guess and i don't know if it is correct .

Thanks !