Ring of subquotients of a finite group I: Linearization

Abstract

We introduce the ring Λ(G) of subquotients of a finite group G.As an abelian group, it is free on the set of conjugacy classes of subquotients of the group G. The ring Λ(G) integrally extends the Burnside ring and there is a linearization map with range the Grothendieck ring of the Mackey algebra.

The Dade group of Mackey functors for p-groups

“…We obtain the result on the rank of the Mackey-Dade group by constructing an isomorphism between this group and the product of all Dade groups of Weyl groups of all subgroups of P . Existence of such an isomorphism also suggests a relation between the Mackey-Dade group, the ring of subquotients of P , and the lineralization map for Mackey functors, both introduced by the author in [10]. We explain this relation in Section 6.…”

Ring of subquotients of a finite group II: Pure bisets

“…In this section we introduce the category of pure bisets. For a review of the terminology of bisets and for our notation, we refer to [6,Section 3.1]. We only recall that given finite groups G and H and a subgroup U of the direct product G × H , we denote the transitive (G, H)-biset with the point stabilizer U by ( G×H U ) and by Bouc's Decomposition Theorem, any such biset is equal to the product of five basic bisets called induction, inflation, isogation, deflation and restriction.…”

“…We only recall that given finite groups G and H and a subgroup U of the direct product G × H , we denote the transitive (G, H)-biset with the point stabilizer U by ( G×H U ) and by Bouc's Decomposition Theorem, any such biset is equal to the product of five basic bisets called induction, inflation, isogation, deflation and restriction. In the notation of [6], we have…”

“…Note that the mark morphism is not compatible with the •-biset functor structure of the functor Λ described in [6]. However there is an alternative definition, described below, of a mark of subquotient on another one which is also compatible with this structure.…”

“…In [6], we introduced the ring Λ(G) of subquotients of a finite group G. As an abelian group, it is free on the set of conjugacy classes of subquotients of G, where by a subquotient we mean a quotient of a subgroup of G. More precisely, we define Similarly, we define the ghost ring Λ * (G) of This extends the definition of the mark of a subgroup on another one. We show that the table of generalized marks is lower triangular with non-zero determinant.…”