Research Council Backup mandate Leo Margolis: Units and Isomorphisms of Group Rings

The goal of the project is to deepen the understanding of the
algebraic structure of group rings by making progress on main open
problems about this structure, developing relevant tools to study it
and opening new horizons to the area by also including new
questions. Roughly the work can be divided into two parts:
Isomorphism Questions and Unit Questions.
Regarding Isomorphism Questions the main open problem of the
area, the Modular Isomorphism Problem, will be studied in depth and
an answer to the problem for several relevant new classes of pgroups
is foreseen, as well as the development of new grouptheoretical
invariants. Moreover the study of a variation of the
Isomorphism
Problem for twisted group rings will be developed,
which
will be of interest to people working with group cohomology
and
extensions of groups.
The study of units in group rings will center on units in the integral
group ring ZG of a finite group G. After the recent solution of the main
question regarding units of finite order in ZG, an analysis will be
undertaken whether a problem posed by Kimmerle can be solved for
solvable groups and serve as the new strong conjecture in the area.
The Prime Graph Question, which makes a statement on the orders
of units in ZG and reduces to almost simple groups, will be further
studied using new techniques for groups of Lie type. Moreover the
study of the abelianization of the unit group will involve techniques
from the theory of infinite groups.