Strongly Lie nilpotent group algebras of index at most 8

Abstract

Let K be a field of characteristic p > 0 and let G be an arbitrary group. In this paper, we classify group algebras KG which are strongly Lie nilpotent of index at most 8. We also show that for k ≤ 6, KG is strongly Lie nilpotent of index k if and only if it is Lie nilpotent of index k.

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

“…If t L (KG) = 10 = 4p − 2, then t L (KG) = 10 by [13]. Also t L (KG) ≤ 8 yields t L (KG) = t L (KG) ≤ 8, by [8,17]. So t L (KG) = 12.…”

“…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] .…”

“…Also t L (KG) = 9, 10 and 11 by [13] and Lemma 2.4. For t L (KG) ≤ 8 yields t L (KG) = t L (KG) ≤ 8, by [8,17]. If t L (KG) = 12, then t L (KG) = 12 by Theorem 2.1.…”

“…Again in all these cases t L (KG) = t L (KG). 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

“…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. Recently, we have some results on classification of Lie nilpotent group algebras of Lie nilpotency index upto 14 (see [2,8,21,23]). Furthermore, group algebras with minimal Lie nilpotency index p + 1 have been classified by Sharma and Bist [20].…”