Groups with 3-Abelian normal closures
By a theorem of \textit{F. W. Levi} [J. Indian Math. Soc., New Ser. 6, 87-97 (1942; Zbl 0061.026)] a group G satisfies the second Engel condition if and only if every element of G is contained in an abelian normal subgroup of G. Call G n-abelian (n-Levi resp. n-Bell) if it satisfies the law \((XY)^ n=X^ nY^ n\) \(([X^ n,Y]=[X,Y]^ n\) resp. \([X^ n,Y]=[X,Y^ n]).\) The following generalization of Levi's result is proved: Theorem 1. For any group G the following are equivalent: i) every element of G is contained in a 3-abelian normal subgroup, ii) G is a 3-Levi group, iii) G is a 3-Bell group, iv) G is a 3-Engel group and \([X,Y,Y]^ 3=1\) is a law in G. - In Theorem 2, some further properties of 3-Bell groups G are given. For example, \((G_ 3)^ 9=(G_ 4)^ 3=G_ 5=1\).
scientific article; zbMATH DE number 4065281