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“Biset functors and the
Tornehave morphism”

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LAURENCE BARKER

(Bilkent University)

ABSTRACT: The usual way of constructing a permutation module
from a permutation set is linearization: one regards the permutation set as a
basis for the permutation module. One can bring in some ring-theoretic
structure by introducing virtual permutation modules and virtual permutation
sets. One can bring in a lot more structure using the notion of a biset functor, which provides a unified way of dealing with
reduction to subgroups and reduction to quotient groups. Tornehave
found a way of constructing modules from precisely those virtual permutation
sets that are killed by linearization. We shall introduce a refinement of his construction,
expressing it as a morphism of biset
functors, and we shall discuss a uniqueness property and an application of this
morphism.Arithmetical investigations led me to ask: For
which finite groups G is the group ring of G over a field K isomorphic to a direct product of
division rings? There is a complete group-theoretical answer to this question,
having been worked out in collaboration with B. Huppert. But ultimately it
leads to a delicate number-theoretical problem of elementary character.

Date : September 27, 2007 (Thursday)

Time : 14.40 The
second hour, 15:40 - 16:30, will be open to all, but more specialized in content.

Place: Mathematics Seminar Room SA141