Lie solvable group algebras of derived length three

Abstract

Let K be a field of characteristic p > 2 and let G be a group. Necessary and sufficient conditions are obtained so that the group algebra KG is strongly Lie solvable of derived length at most 3. It is also shown that these conditions are equivalent to KG Lie solvable of derived length 3 in characteristic p ≥ 7.

On group algebras with unit groups of derived length three\\ in characteristic three

“…As before, on replacing g by x r g we get Conversely, if G is a central and elementary abelian 3-subgroup of G of order ≤ 3 3 then by [12,Theorem 2.3], we have δ (3) (KG) = 0 and hence U = 1.…”

Lie Solvable Group Algebras of Derived Length Three in Characteristic Three

“…Conversely, it is obvious by [11,Theorem 2.3], that if G is a group such that G = C 3 or G = C 3 × C 3 and x g = x −1 for all x ∈ G , g / ∈ C G (G ), then KG is strongly Lie solvable of derived length 3 and hence δ [3] (L(KG)) = 0. Theorem 2.7.…”

Group Algebras with Minimal Strong Lie Derived Length

“…Actually, F. Levin and G. Rosenberger characterized Lie metabelian modular group algebras and showed that this condition is equivalent to saying that the group algebra is strongly Lie metabelian [3]. Moreover, M. Sahai [7] classified group algebras over fields of odd characteristic whose strong Lie derived length is at most 3. Thus they already completed the special cases p = 2, 3, 5, 7.…”