Group algebras with almost minimal Lie nilpotency index

Abstract

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.

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

“…So r = 1, G ′ ∼ = C 9 × C 3 and |γ 3 (G)G ′3 | = 9. Thus either γ 3 [13] , a contradiction. [13] = 0, a contradiction.…”

“…If γ 4 (G) ∼ = C 3 , then as in [3] by using [2], there exist x, y ∈ G such that a = (x, y), b = (x, y, y), c = (x, y, y, y), G ′ = a, b, c , γ 3 (G) = b, c and γ 4 (G) = c . Therefore by Lemmas 2.1 and 2.2, (a−1) 2 (b−1) 2 (c −1) 2 ∈ KG [13] = 0, a contradiction. Proof.…”

“…If r = 1 and G ′ ∼ = (C 3 ) 5 , then γ 3 (G) ∼ = C 3 . But then by [13,Lemma 2.7], t L (KG) = 14. So r = 1, G ′ ∼ = C 9 × C 3 and |γ 3 (G)G ′3 | = 9.…”

“…In [21], it is proved that t L (KG) = p + 1 if and only if t L (KG) = p + 1. Lie nilpotent group algebras with Lie nilpotency index 2p, 3p − 1 or 4p − 2 have been investigated in [13]. Again in all these cases t L (KG) = t L (KG).…”

“…In [18], it is proved that if G is a nilpotent group with |G ′ | = p n , then p + 1 ≤ t L (KG) ≤ t L (KG) ≤ |G ′ | + 1. 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].…”