ZC · Torsion units of integral group rings

Torsion units of integral group rings

The problem studied in this project was the so-called Zassenhaus Conjecture. Hans Julius Zassenhaus (1912-1991) made important contributions to algebra. Group rings were introduced in the late 19th century as a tool to study symmetries, matrices and other algebraic objects. The group ring RG of a group G over a ring R can be defined as a vector space over R where the base is given by the elements of the group G. This provides an additive structure and the multiplicative structure on RG is given by extending the multiplications on G and R and declaring elements of G and R to commute. From the middle of the 20th century the group ring became an object of interest in itself. It can be seen as a structure joining in an elegant manner the algebraic theories on rings and group. When R=Z, the ring of integers number theory enters the picture and makes the closest connection to the group base G, since it keeps the arithmetic information which would be lost when one is allowed to divide by some primes. The first to study particularly the unit group of the integral group ring was Graham Higman in his PhD thesis in 1940 who proved that for abelian G the units of finite order in ZG are, up to sign, exactly the elements of the group base G. This gave support to a conjecture that was probably mentioned by the specialist during that years and that was publised formally by Zassenhaus in 1974. This conjecture stated that if G is a finite group then a unit of finite order in ZG should be should be conjugate in the rational group algebra, up to sign, to an element of the group base. This conjecture inspired many research carried out during the following decade and it became one of the central problems in the study of integral group rings. The impact of mathematics on society is manifold and often comes at a later stage in a very surprising manner. For example the ideal of groups rings now play a role in cryptography, but were originally studied as an object of pure scientific interest. It is hence impossible to predict, as mostly with fundamental research, which could be the consequences and practically useful implementations of this research. Concretely the objectives of this project were to study the Zassenhaus Conjeture for several new classes of groups, which were more concretely the class of metabelian groups and the projective special linear groups PSL(2,q). Also the project included the study of a weaker version of the Zassenhaus Conjeture, the so-called Prime Graph Question, introduced by Wolfgang Kimmerle in 2006.

Текст от CORDIS, на английски · Данни: CORDIS, © Европейски съюз