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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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