Let $(G,\cdot)$ be a group show that
A) $G$ is abelian
B) For all $x,y\in G: (xy)^{-1}=x^{-1}y^{-1}$
C) For all $x,y\in G: (xy)^{2}=x^{2}y^{2}$
D) There existst an $n\in \mathbb{Z}$ such that for all $x,y\in G$
$\displaystyle (xy)^{n-1}=x^{n-1}y^{n-1}$ and $\displaystyle (xy)^{n}=x^{n}y^{n}$ and $\displaystyle (xy)^{n+1}=x^{n+1}y^{n+1}$
are equivalent.
I showed $A\iff B,A\iff C$ and $A\iff D$. That was not really hard, except maybe $D\Rightarrow A$. I would like to do a circular argument, but i have no idea how. Any hints?
Thanks in advance!
So you're basically asking how to prove that B)$\implies$C) and C)$\implies$D).
$\bbox[5px,border:2px solid #4B0082]{B)\implies C)}$
Note that $$\begin{align} \forall x,y\in G\left((xy)^{-1}=x^{-1}y^{-1}\right) &\implies \forall x,y\in G\left(y^{-1}x^{-1}=x^{-1}y^{-1}\right)\\ &\implies \forall x,y\in G\left(xy=yx\right)\\ &\implies \forall x,y\in G\left((xy)^2=x^2y^2\right).\end{align}$$
$\bbox[5px,border:2px solid #4B0082]{C)\implies D)}$
As noted by Seth in the comments, assume C) is true and let $n=1$.