On \(\mathbb Z_6\)-graded Lie rings
\textit{V. A. Kreknin} [Sov. Math., Dokl. 4, 683-685 (1963); translation from Dokl. Akad. Nauk SSSR 150, 467--469 (1963; Zbl 0134.03604)] showed that a \(\mathbb Z/(n)\)-graded Lie ring \[ \mathfrak g=\bigoplus_ {i\in\mathbb Z/(n)}\mathfrak g_ i \] with trivial zero component is solvable of derived length \(\leq2^ {n-1}-1\). It is conjectured that the derived length be bounded by \(n-1\). Here the case \(n=6\) is studied by the authors thus completing work of Higman and Khukhro for \(n=2,3,4,5,7\). The following theorem is proved: Let \(\mathfrak g\) be a \(\mathbb Z/(6)\)-graded Lie ring with \(\mathfrak g_ 0=(0)\). Then \(\mathfrak g\) is solvable of derived length at most 4, and the third constituent \(\gamma_ 3(\mathfrak g)\) of the lower central series of \(\mathfrak g\) is nilpotent of nilpotency class at most 6. See also the authors work with \textit{L. Wilson} in J. Algebra 283, No. 1, 149--160 (2005; Zbl 1058.17010 ).
scientific article; zbMATH DE number 2105248