Let K be a field of characteristic p > 0 and let G be an arbitrary non-abelian group. It is well known that if KG is Lie nilpotent, then its upper as well as lower Lie nilpotency index is at least p + 1. Shalev investigated Lie nilpotent group algebras whose Lie nilpotency indices are next lower, namely 2p and 3p − 1 for p ≥ 5 and obtained certain interesting results. The aim of this paper is to classify group algebras KG which are Lie nilpotent having Lie nilpotency indices 2p, 3p − 1 and 4p − 2. Our proofs are independent of Shalev and are valid for p = 2 and 3 as well.

In this paper, we classify the modular group algebra KG of a group G over a field K of characteristic p > 0 having upper Lie nilpotency index t L (KG) = |G ′ | − k(p − 1) + 1 for k = 14 and 15. Group algebras of upper Lie nilpotency index |G ′ | − k(p − 1) + 1 for k ≤ 13, have already been characterized completely.

In this paper, we establish the structure of the unit group of the group algebra [Formula: see text] where [Formula: see text] is an abelian group of order at most 16 and [Formula: see text] is a finite field of characteristic [Formula: see text] with [Formula: see text] elements.

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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.