The nilpotency class of the unit group of a modular group algebra I

The nilpotency class of the unit groupU of a modularp-group algebraFG is determined whenp is odd andG has a cyclic commutator subgroup. This is done via an extension of a theorem of Coleman and Passman, dealing with wreath products obtained as sections ofU.     

On the nilpotency class of a multiplicative group of a modular group algebra of a dihedral 2-group

It is proved that the wreath product of a second-order group and the commutant of a dihedral group is imbedded into a multiplicative group of a modular group algebra of a dihedral group of order 2 ⁿ . This implies that the nilpotency class of the multiplicative group is equal to 2 ^n–2, i.e., to the order of the commutant of the dihedral group.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 1, pp. 39–45, January, 1995.     

The Auslander-Reiten quiver of a modular group algebra revisited

This work has been supported by the DFG project “Darstellungstheorie” at the University of Essen.     

The modular group algebras of p-groups of maximal class II

The modular isomorphism problem is settled for 3-groups of maximal class but two families of groups. Moreover, the conjecture that the ideals belonging to the lower central series of a group base are determined by the structure of the group algebra is refuted in greatest generality by virtue of a single group of order 81 of maximal class, and it is proved that the nilpotency class is determined by the structure of the group algebra for p-groups of maximal class.     

The Lie structure on the Hochschild cohomology of a modular group algebra

We prove that the Gerstenhaber bracket on the Hochschild cohomology of the group algebra of a cyclic group over a field of positive characteristic is not trivial. In this case, we relate the Lie algebra structure on the odd degrees of the Hochschild cohomology with a Witt-type algebra.     

The hochschild cohomology ring of a modular group algebra: the commutative case

Let k be a field of characteristic P >0 and let G be a finite abelian group. We determine the structure of the Hochschild cohomology ring of the group algebra k G. Moreover, we prove that for any finite group G the Krull dimension of H H ^*(k G) equals the p-rank of G.     

On nilpotency of the barideal of a bernstein algebra

It is given a new bound for the index of nilpotency of the barideal of a finited generated Bernstein algebra. Also, it is found the index of solvability of the barideal of a Bernstein algebra and a bound of nilpotency for the square of the barideal.     

On the order of the unitary subgroup of a modular group algebra

Let KG be a group algebra of a finite p-group G over a finite field Kof characteristic p. We compute the order of the unitary subgroup of the group of units when G is either an extraspecial 2-group or the central product of such a group with a cyclic group of order 4 or G has an abelian subgroup A of index 2 and an element b such that b inverts each element of A.     

Free groups and involutions in the unit group of a group algebra

We study the existence of free groups of rank 2 in the group generated by the involutions in the unit group of a group algebra over a non-absolute field.Received: 25 March 2004     

Simplicity condition for the fundamental ideal of the modular group algebra of a locally finite group

A criterion for the simplicity of the fundamental ideal of the group ring of a locally finite group is given. With its help the simplicity of this ideal for the group PSL(2, F), where F is an infinite field that is a union of its finite subfields, is proved.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, Nos. 7 and 8, pp. 1088–1091, July–August, 1991.     

On the unit group of a semisimple group algebra and the normal complement problem

Archiv der Mathematik - Let F be the field with p elements, where p is of the form $$ (2t+1)$$ for some square free odd integer t. In this article, we obtain the order of the symmetric and the...     

Generalized nilpotency of the multiplicative group of a group ring

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 9, pp. 1179–1183, September, 1989.