Let $n$ be an integer $\geqslant 2$. A group $G$ is called generalized
$n$-abelian if it admits a {\em positive polynomial} endomorphism
of degree $n$, that is if there exist $n$ elements $a_1, a_2,
\dots, a_n$ of $G$ such that the function $\varphi: x\mapsto
x^{a_1}x^{a_2}\cdots x^{a_n}$ is an endomorphism of $G$. In this
paper we give some sufficient conditions for a generalized
$n$-abelian group to be abelian. In particular, we show that
every group admitting a positive polynomial monomorphism of
degree 3 is abelian.