Modular group algebras with centrally metabelian unit groups

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

On monomial codes in modular group algebras

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] for [Formula: see text] and [Formula: see text]. Group algebras of upper Lie nilpotency index [Formula: see text] for [Formula: see text], have already been characterized completely.

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

p ∗-Theory and modular representation theory

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.

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 𝐹 be a field of odd characteristic and 𝐺 a group for which the group algebra F ⁢ G FG is modular. In this paper, necessary and sufficient conditions are presented for the commutator subgroup of the unit group of F ⁢ G FG to be nilpotent of class at most 2.

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).

It is shown that the isomorphism type of a metacyclic p p -group is determined by its group algebra over the field F F of p p elements. This completes work of Bagiński. It is also shown that, if a p p -group G G has a cyclic commutator subgroup G ′ G’ , then the order of the largest cyclic subgroup containing G ′ G’ is determined by F G FG .

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.

We study the Modular Isomorphism Problem applying a combination of existing and new techniques. We make use of the small group algebra to give a positive answer for two classes of groups of nilpotency class 3. We also introduce a new approach to derive properties of the lower central series of a finite 𝑝-group from the structure of the associated modular group algebra. Finally, we study the class of so-called 𝑝-obelisks which are highlighted by recent computer-aided investigations of the problem.