Endomorphism and abelian group

Let $(\mathcal{G}, \cdot)$ be a group for which there exist an endomorphism $f:\mathcal{G} \rightarrow \mathcal{G}$ such that

$$f\left ( x^n y^{n+1} \right )= x^{n+1}y^n \quad \text{forall} \; x, y \in \mathcal{G}$$

Prove that $\mathcal{G}$ is abelian.

Solution

Setting $x=y^{-1}$ we get that $f(y)=y^{-1}$. Therefore, forall $x, y \in \mathcal{G}$ , since $f$ is an endomorphism we have that:

\begin{align*}
xy &=\left ( y^{-1} x^{-1} \right )^{-1} \\
 &= f\left ( y^{-1} x^{-1} \right )\\
 &= f\left ( y^{-1} \right ) f \left ( x^{-1} \right ) \\
 &= yx
\end{align*}

Thus $\mathcal{G}$ is abelian.