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Lie nilpotency indices of modular group algebras. (English) Zbl 1197.16030
Let $$R$$ be an associative algebra with identity. View $$R$$ as a Lie algebra under commutation: $$[a,b]=ab-ba$$. Define the series $$\{R^{(n)}\}_{n\geq 1}$$ (resp. $$\{R^{[n]}\}_{n\geq 1}$$) of two-sided ideals in $$R$$ by setting $$R^{[1]}=R^{(1)}=R$$ and for $$n\geq 0$$, $$R^{(n+1)}$$ to be the two-sided ideal of $$R$$ generated by all elements $$[a,b]$$ with $$a\in R$$, $$b\in R^{(n)}$$ (resp. $$R^{[n+1]}$$ to be the two-sided ideal generated by all left normed Lie commutators $$[a_1,a_2,\dots,a_{n+1}]$$.)
Let $$KG$$ be the group algebra of a group $$G$$ over a field $$K$$ of positive characteristic. Let $$t^L(KG)$$ (resp. $$t_L(KG)$$) denote the least integer $$n$$, if it exists, such that $$KG^{(n)}=0$$ (resp. $$KG^{[n]}=0$$). It is known that $$KG$$ is Lie nilpotent, i.e., $$KG^{[n]}=0$$ for some integer $$n$$, if and only if $$KG^{(m)}=0$$ for some integer $$m$$, and $$t_L(KG)\leq t^L(KG)\leq |G'|+1$$, where $$G'$$ is the derived subgroup of $$G$$. The precise value of $$t^L(KG)$$ can be given in terms of the indices $$[D_{(n+1)}(G):D_{(n)}(G)]$$, where $$D_{(i)}(G)=G\cap (1+KG^{(i)})$$, $$i\geq 1$$.
Continuing the work in [V. Bovdi, Sci. Math. Jpn. 65, No. 2, 267-271 (2007; Zbl 1130.16016), V. Bovdi, T. Juhász, E. Spinelli, Algebr. Represent. Theory 9, No. 3, 259-266 (2006; Zbl 1115.16012), V. Bovdi, E. Spinelli, Publ. Math. 65, No. 1-2, 243-252 (2004; Zbl 1070.16026), A. Shalev, Arch. Math. 60, No. 2, 136-145 (1993; Zbl 0818.16025)], where the class of groups $$G$$ for which the indices $$t^L(KG)$$ ($$t_L(KG)$$) are maximal or almost maximal has been characterized, the authors of the present paper characterize the groups $$G$$ for which the Lie nilpotency index $$t^L(KG)$$ (or $$t_L(KG)$$) is the next highest possible.

##### MSC:
 16S34 Group rings 17B30 Solvable, nilpotent (super)algebras 20C07 Group rings of infinite groups and their modules (group-theoretic aspects) 16W10 Rings with involution; Lie, Jordan and other nonassociative structures
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