Lie Nilpotency Indices of Modular Group Algebras

Abstract

Let K be a field of positive characteristic p and KG the group algebra of a group G. It is known that if KG is Lie nilpotent then its upper (or lower) Lie nilpotency index is at most |G ′ | + 1, where |G ′ | is the order of the commutator subgroup. The class of groups G for which these indices are maximal or almost maximal have already been determined. Here we determine G for which upper (or lower) Lie nilpotency index is the next highest possible.1991 Mathematics Subject Classification. 16S34, 17B30.

A note on the upper Lie nilpotency index of a group algebra

“…Let R be an associative ring with unity and let 2 ∈ U(R). Then for all x, y ∈ U(R), ((x, y, y, y) − 1) 2 ∈ R [7] .…”

“…By Lemma 1.1, β ∈ R [5] and by [22, Lemma 2.2], α 2 ∈ R [7] . Also αβ, βα ∈ R [7] and β 2 ∈ R [8] . Hence (t 3 − 1) 2 ∈ R [7] .…”

“…A complete description of Lie nilpotent modular group algebras KG of upper Lie nilpotency index t L (KG) ≤ 9p − 7 is given in [9,10,11,12,17]. On the other hand, group algebras with t L (KG) = |G ′ | − k(p − 1) + 1 for k ≤ 13 are classified in [2,3,5,6,13,14,18]. In this paper, we classify the group algebra KG with t L (KG) = |G ′ | − k(p − 1) + 1 for k = 14 and 15. our terminology and notations are same as in [10].…”

“…A complete description of Lie nilpotent modular group algebras KG with Lie nilpotency index at most 8 is given in [8,17]. More results on Lie nilpotency indices of modular group algebras are given in [3,4,6,7,14,15,16,20].…”

On the Lie nilpotency index of modular group algebras

“…This problem was completed by [6]. Results on the next smaller Lie nilpotency index can be easily seen in [4][5][6][7]. In [3], Bovdi and Kurdics discussed the upper and lower Lie nilpotency index of a modular group algebra of metabelian group G and determine the nilpotency class of the group of units.…”