Abstract
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.
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Daoud, B., Hamitouche, M. & Merikhi, K. On the Nilpotency Class of a Generalized 3-Abelian Group. Mediterr. J. Math. 10, 1189–1194 (2013). https://doi.org/10.1007/s00009-013-0264-2
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DOI: https://doi.org/10.1007/s00009-013-0264-2