Lie nilpotency index of a modular group algebra

Let K be a field of positive characteristic p and KG the group algebra of a group G. It is known that if KG is Lie nilpotent, then its upper (or lower) Lie nilpotency index is at most |G′| + 1, where |G′| is the order of the commutator subgroup. The class of groups G for which these indices are maximal or almost maximal has already been determined. Here we determine G for which upper (or lower) Lie nilpotency index is the next highest possible.

A deformation-theoretic version of Maschke's theorem for modular group algebras: The commutative case

Let [Formula: see text] be a group and let [Formula: see text] be a field of characteristic [Formula: see text]. Lie nilpotent group algebras of strong Lie nilpotency index at most 13 have been classified by many authors. In this paper, our aim is to classify the group algebras [Formula: see text] which are strongly Lie nilpotent of index 14.

Let [Formula: see text] be an arbitrary group and let [Formula: see text] be a field of characteristic [Formula: see text]. In this paper, we give some improvements of the upper bound of the lower Lie nilpotency index [Formula: see text] of the group algebra [Formula: see text]. We also give improved bounds for [Formula: see text], where [Formula: see text] is the number of independent generators of the finite abelian group [Formula: see text]. Furthermore, we give a description of the Lie nilpotent group algebra [Formula: see text] with [Formula: see text] or [Formula: see text]. We also show that for [Formula: see text] and [Formula: see text], [Formula: see text] if and only if [Formula: see text], where [Formula: see text] is the upper Lie nilpotency index of [Formula: see text].

On monomial codes in modular group algebras

Modular group algebras with centrally metabelian unit groups

In this paper, we classify the modular group algebra [Formula: see text] of a group [Formula: see text] over a field [Formula: see text] of characteristic [Formula: see text] having upper Lie nilpotency index [Formula: see text]. The group algebra [Formula: see text] with [Formula: see text] has already been described.

Let p be a prime, R a commutative ring of characteristic p , G a p -primary abelian group, and V(RG) the group of normalized units in the group algebra RG . It is largely determined when V(RG) is totally projective. This generalizes the well-known special case when R is a perfect field. Moreover, the current proof has the advantage of greater clarity.

The Loewy structure of projective indecomposable modules for some modular group algebras

Sections and Irreducible Modules

For modular abelian group algebras over a perfect field of prime characteristic, a connection is given between the direct factor problem and the isomorphism problem.

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.

The Jacobson Radical of Group Algebras

We prove certain results pertaining to some isomorphism properties of commutative modular group algebras and briefly review a paper by pointing out some obvious mistakes and essential incorrectness.

We study the unitary subgroupV *(F2 G) in the group algebras F2 Gof 2-groups of maximal class over the field F2of two elements. We show that there does not exist a normal complement to Gin V *(F2 G) if and only if Gis the dihedral group of order 2n(n≥ 5) or the semidihedral group of order 2n(n≥5). We also describe V *(F2 G) when the subgroup Ghas order 8 and 16 and in this case there exist a normal complement of Gin V *(F2 G).