The derived length of Lie soluble group rings. I (Q1196782)

Let the group algebra \(KG\) of a group \(G\) over a field \(K\) be considered as a Lie algebra under commutation. The precise conditions under which \(KG\) is Lie soluble are known [\textit{I. B. S. Passi}, \textit{D. S. Passman} and \textit{S. K. Sehgal}, Can. J. Math. 25, 748-757 (1973; Zbl 0266.16011)]. In characteristic zero this is the case if and only if \(G\) is Abelian. This raises the problem of relating the structure of \(G\) to \(d\ell(KG)\) the derived length of \(KG\). The present paper is a contribution in this direction. The main result (Theorem A) states that if \(K\) is a field of characteristic \(p>0\) and \(G\) is a non-abelian group, then \(d\ell(KG)\geq\lceil\log_2(p+1)\rceil\), where \(\lceil r\rceil\) denotes the upper integral part of a real number \(r\). Furthermore, it is proved (Proposition B) that equality holds if the commutator subgroup of \(G\) is central of order \(p\).