Lie properties of the group algebra and the nilpotency class of the group of units
Let \(\mathbb{F}\) be a field of characteristic \(p\) and let \(G\) be a group having a nontrivial Sylow \(p\)-subgroup. Then the group of units \(U(\mathbb{F}[G])\) of the group algebra \(\mathbb{F}[G]\) is nilpotent if and only if \(G\) is nilpotent and the commutator subgroup \(G'\) is a group of prime power order [\textit{I. I. Khripta}, Mat. Zametki 11, 191-200 (1972; Zbl 0258.20005)]. This paper is aimed at making a contribution to the determination of the nilpotency class of \(U(\mathbb{F}[G])\). If \(t_L(\mathbb{F}[G])\) and \(t^L(\mathbb{F}[G])\) denote respectively the lower and the upper Lie nilpotency indices of \(\mathbb{F}[G]\), then, in view of the result of \textit{N. Gupta} and \textit{F. Levin} [J. Algebra 81, 225-231 (1983; Zbl 0514.16024)], \(\text{cl}(U(\mathbb{F}[G]))+1\leq t_L(\mathbb{F}[G])\leq t^L(\mathbb{F}[G])\). If \(p>3\), then, by a result of \textit{A. Bhandari} and the reviewer [Bull. Lond. Math. Soc. 24, No. 1, 68-70 (1992; Zbl 0777.20004)], \(t_L(\mathbb{F}[G])=t^L(\mathbb{F}[G])\); using this result \textit{A. Shalev} [Arch. Math. 60, No. 2, 136-145 (1993; Zbl 0818.16025)] determined the Lie nilpotency indices of \(\mathbb{F}[G]\), and consequently \(\text{cl}(U(\mathbb{F}[G]))\), for certain groups. The authors in this paper, extending some of the results of Shalev, determine \(t_L(\mathbb{F}[G])\), \(t^L(\mathbb{F}[G])\), and \(\text{cl}(U(\mathbb{F}[G]))\) for certain metabelian groups. Among other results it is proved that if \(G\) is a nilpotent group with a cyclic commutator subgroup \(G'\) of order \(p^n>2\) and \(\mathbb{F}\) a field of prime characteristic \(p\), then \(U(\mathbb{F}[G])\) is nilpotent of class \(p^n-1\) if \(\text{Syl}_p(G)=G'\), and of class \(p^n\) if \(\text{Syl}_p(G)\neq G'\). The paper concludes with the description of modular group algebras with unit group nilpotent of class 3, thus extending the work of \textit{M. A. Rao} and \textit{R. Sanding} [Can. Math. Bull. 38, No. 1, 112-116 (1995; Zbl 0824.20006)] on modular group algebras of finite \(p\)-groups.
scientific article; zbMATH DE number 1271643
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