On the Nilpotency Class of a Generalized 3-Abelian Group

A group G is called 3-abelian if the map ${x mapsto x^{3}}$ is an endomorphism of G and it is called generalized 3-abelian, if there exist elements ${c_{1}, c_{2}, c_{3} in G}$ such that the map ${varphi : x longmapsto {x^{c_{1}} x^{c_{2}} x^{c_{3}}}}$ is an endomorphism of G. Abdollahi, Daoud and Endimioni have proved that a generalized 3-abelian group G is nilpotent of class at most 10. Here, we improve the bound to 3 and we show that the exponent of its derived subgoup is finite and divides 9. We also prove that G is 3-Levi, 9-central, 9-abelian and 3-nilpotent of class at most 2.