Browsing by Subject "Burnside problem."

Now showing 1 - 5 of 5

Open Access
A correspondence of simple alcahestic group functors
(2008) Coşkun, Olcay
Representation theory of finite groups associates two classical constructions to a group G, namely the representation ring of G and the Burnside ring of G. These rings share a special structure that comes from three classical maps, namely restriction, conjugation, and transfer maps. These are not the only objects having this structure and the theory of Mackey functors, introduced by Green, unifies the treatment of such objects. The above constructions share a further structure that comes from two other maps, the inflation map and the deflation map. Unified treatment of the objects having this further structure was introduced by Bouc [4]. These objects are called biset functors. Between Mackey functors and biset functors there lies more natural constructions, for example the functor of group (co)homology. In order to handle these intermediate structures, Bouc introduced another concept, now known as globallydefined Mackey functors, a name given by Webb. In this thesis, we unify the above theories by considering the algebra whose module category is equivalent to the category of biset functors and by introducing alcahestic group functors. Our main results classify and describe simple alcahestic group functors and give a criterion of semisimplicity for the categories of these functors.
Open Access
Modular representations and monomial burnside rings
(2004) Coşkun, Olcay
We introduce canonical induction formulae for some character rings of a finite group, some of which follows from the formula for the complex character ring constructed by Boltje. The rings we will investigate are the ring of modular characters, the ring of characters over a number field, in particular, the field of real numbers and the ring of rational characters of a finite p−group. We also find the image of primitive idempotents of the algebra of the complex and modular character rings under the corresponding canonical induction formulae. The thesis also contains a summary of the theory of the canonical induction formula and a review of the induction theorems that are used to construct the formulae mentioned above.
Open Access
The monomial Burnside functor
(2009) Okay, Cihan
Given a finite group G, we can realize the permutation modules by the linearization map defined from the Burnside ring B(G) to the character ring of G, denoted AK(G). But not all KG-modules are permutation modules. To realize all the KGmodules we need to replace B(G) by the monomial Burnside ring BC(G). We can get information about monomial Burnside ring of G by considering subgroups or quotient groups of G. For this the setting of biset functors is suitable. We can consider the monomial Burnside ring as a biset functor and study the elemental maps: transfer, retriction, inflation, deflation and isogation. Among these maps, deflation is somewhat difficult and requires more consideration. In particular, we examine deflation for p-groups and study the simple composition factors of the monomial Burnside functor for 2-groups with the fibre group {±1}.
Open Access
On the exponential map of the Burnside ring
(2002) Yaman, Ayşe
We study the exponential map of the Burnside ring. We prove the equivalence of the three different characterizations of this map and examine the surjectivity in order to describe the elements of the unit group of the Burnside ring more explicitly.
Open Access
Real monomial Burnside rings and a decomposition of the the tom Dieck map
(2009) Tuvay, İpek
This thesis is mainly concerned with a decomposition of the reduced tom Dieck map die : f A(RG) → B(G) × into two maps die+ and die− of the real monomial Burnside ring. The key idea is to introduce a real Lefschetz invariant as an element of the real monomial Burnside ring and to generalize the assertion that the image of an RG-module under the tom Dieck map coincides with the Lefschetz invariant of the sphere of the same module.