On Lie nilpotent modular group algebras

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

“…Thus γ 6 (G) = 1 and γ 3 (G) is an abelian group. Also |G ′ | ≥ 3 3 and the exponent of γ 3 (G) is at most 3, as for a 1 , a 2 , a 3 ∈ G, ((a 1 , a 2 , a 3 ) −1) 8 ∈ (KG [3] ) 8 ⊆ KG [14] . By Lemma 2.3, (3 m i − 1) is an even number and γ 3 (G)G ′3 /G ′3 = 3 r .…”

“…Let m 2 = 2, m 3 = 1. Then by [3], there exist x, y, z ∈ G such that b 1 = (x, y), b 2 = (x, z), c 1 = (x, y, y), G ′ = b 1 , b 2 , c 1 , d 1 , γ 3 (G) = c 1 and γ 4 (G) = d 1 Therefore by Lemma 2.1, (b 1 − 1) 2 (b 2 − 1) 2 (c 1 − 1) 2 (d 1 − 1) 2 ∈ KG[14] = 0, a contradiction. Suppose m 2 = m 3 = 1, then as in Lemma 2.5, G ′ is abelian and |G ′ | = 3 3 .…”

“…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].…”

“…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].…”

“…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]. Throughout this paper, S(n, m) denotes the group number m of order n from the Small Groups Library-GAP [7].…”