Strongly Lie solvable group algebras of derived length 4 (Q330658)

Let \(KG\) be the group algebra of the group \(G\) over the field \(K\) of prime characteristic \(p\), \(L(KG)\) its associated Lie algebra. The derived length \(d\) of soluble \(L(KG)\) has been investigated recently by many scholars such as \textit{H. Chandra} and the third author [J. Algebra Appl. 11, No. 5, 1250098, 12 p. (2012; Zbl 1262.16020)] describing \(G\) when \(G\) is assumed to be metabelian non-2-Engel, \(d=3\) and \(p=3\). The main result of the present paper is as follows: Let \(p>5\). Then \(d\leq4\) if and only if \(G\) is one of the following types: {\parindent=7mm \begin{itemize}\item[(1)] abelian; \item[(2)] \(G'=C_{13}\) and \(\gamma_3(G)=1\) provided \(p=13\); \item[(3)] \(G'=C_{11}\) and \(\gamma_3(G)=1\) provided \(p=11\); \item[(4)] either \(G'=C_{7}\), or \(G'=C_7\times C_7\) and \(\gamma_3(G)=1\) provided \(p=7\). \end{itemize}}