Michael Dokuchaev

Globalizations of partial actions on nonunital rings

In this note we prove a criteria for the existence of a globalization for a given partial action of a group on an s-unital ring. If the globalization exists, it is unique in a natural sense. This extends the globalization theorem from Dokuchaev and Exel, 2005, obtained in the context of rings with 1.

TWISTED PARTIAL ACTIONS OF HOPF ALGEBRAS

In this work, the notion of a twisted partial Hopf action is introduced as a unified approach for twisted partial group actions, partial Hopf actions and twisted actions of Hopf algebras. The conditions on partial cocycles are established in order to construct partial crossed products, which are also related to partially cleft extensions of algebras. Examples are elaborated using algebraic groups.

Integral group rings of frobenius groups and the conjectures of H.J. Zassenhaus

The conjecture of H.J. Zassenhaus for finite subgroups of units of integral group rings. restricted to p-subgroups, is proved for finite Frobenius groups when p is an odd prime. The result for 2-subgroups is established for those Frobenius groups that cannot be mapped homo-morphically onto S

GLOBALIZATION OF TWISTED PARTIAL HOPF ACTIONS

sub:5

UNITS IN GROUP RINGS OF FREE PRODUCTS OF PRIME CYCLIC GROUPS

esub:. The conjecture in its full strength is proved for A5, S5 and SL(2.5).

Partial actions and Galois theory

In this work, we review some properties of twisted partial actions of Hopf algebras on unital algebras and give necessary and sufficient conditions for a twisted partial action to have a globalization. We also elaborate a series of examples.

Partial Representations and Partial Group Algebras

Let G be a free product of cyclic groups of prime order. The structu re of the unit groupU( G) of the rational group ring G is given in terms of free products and amalgamated free products of groups. As an application, all finite subgroups ofU( G), up to conjugacy, are described and the Zassenhaus Conjecture for finite subgroups in G is proved. A strong version of the Tits Alternative for U( G) is obtained as a corollary of the structural result.