Lie centrally metabelian group rings

Group algebras with centrally metabelian unit groups

“…The characterization of metabelian group algebras by Rosenberger and Levin [2] shows that for a finite group G and K a field with Char K = 2, U (KG) is metabelian if and only if the group algebra KG is Lie metabelian. Also, in this connection, we have an important result due to Sharma and Srivastava [6,Theorem 4.1], which is, δ 2 (U (R))−1 ⊆ δ 2 (L(R))R for arbitrary rings R. This shows [6,Corollary 4.2] that the unit group of a Lie metabelian ring is a metabelian group.…”

“…Recall that a group G is centrally metabelian if the second derived term δ 2 (G) is contained in the centre ζ (G), that is, (δ 2 (G), G) = 1. Recently Sharma and Srivastava [6] and Sahai and Srivastava [4] have obtained necessary and sufficient conditions for the group algebra KG to be Lie centrally metabelian. Our investigations show that U (KG) is centrally metabelian as a group if and only if KG is Lie centrally metabelian, at least when Char K = p = 2 and G is a finite group.…”

“…This gives t g = t. Thus t g = t for all g ∈ G and so G = t is central in G. Now by using [2] and Theorem 2.1, we get that statements (i), (ii) and (iv) given above are equivalent. Further by [6,Corollary 4.2], KG Lie metabelian implies U (KG) is metabelian. Now [5,Theorem B] gives that either G is abelian or G is nilpotent with…”

Lie Centre-by-Metabelian Group Algebras over Commutative Rings

“…The classification for the case where k is a field of characteristic p > 0 is distributed among several articles by Külshammer, Sahai, Sharma, Srivastava, and myself [3,[5][6][7][8]10]. The results found there carry over to the case where k is a ring of prime characteristic p, since p ⊆ k, and is bilinear.…”

Lie solvable group algebras of derived length three

“…In this direction Levin and Rosenberger [1] have characterized Lie metabelian group rings. Lie centrally metabelian group algebras have been studied by Sharma and Srivastava [6] and Sahai and Srivastava [3]. It is shown in [1] that the group ring RG of a group G over a commutative ring R is Lie metabelian if and only if it is strongly Lie metabelian.…”