Biset functors for finite groups by serge Bouc
By serge Bouc
This quantity exposes the speculation of biset functors for finite teams, which yields a unified framework for operations of induction, limit, inflation, deflation and shipping via isomorphism. the 1st half remembers the fundamentals on biset different types and biset functors. the second one half is worried with the Burnside functor and the functor of complicated characters, including semisimplicity matters and an outline of eco-friendly biset functors. The final half is dedicated to biset functors outlined over p-groups for a set leading quantity p. This contains the constitution of the functor of rational representations and rational p-biset functors. The final chapters reveal 3 functions of biset functors to long-standing open difficulties, particularly the constitution of the Dade staff of an arbitrary finite p-group.This ebook is meant either to scholars and researchers, because it provides a didactic exposition of the fundamentals and a rewriting of complex ends up in the world, with a few new principles and proofs.
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Extra info for Biset functors for finite groups
6. Corollary : Let X and Y be ﬁnite G-sets. Then [X] and [Y ] have the same image in B(G) if and only if X and Y are isomorphic. Proof: Indeed [X] and [Y ] have the same image in B(G) if and only if there exist positive integers m ≤ n, ﬁnite G-sets Zi and Ti , for i = 1, . . n, and an isomorphism of G-sets m X n (Zi Ti ) ( i=1 n Zi ) ( i=m+1 Ti ) ∼ =Y i=m+1 m ( i=1 m Zi ) ( i=1 n Ti ) (Zi Ti ) . i=m+1 Counting ﬁxed points on each side by a subgroup H of G shows that |X H | = |Y H |. Since this holds for any H, the G-sets X and Y are isomorphic.
2. Remark (presentation of the biset category) : It follows from this deﬁnition that the category C is a preadditive category, in the sense of Mac Lane ( I Sect 8): the sets of morphisms in C are abelian groups, and the composition of morphisms is bilinear. If G and H are ﬁnite groups, then any morphism from G to H in C is a linear combination with integral coeﬃcients of morphisms of the form [(H × G)/L], where L is some subgroup of H × G. 26, any such morphism factors in C as the composition G ResG B GB Def B B/A G B/A Iso(f ) G D/C Inf D D/C GD IndH D GH, for suitable sections (B, A) and (D, C) of G and H respectively, and a group isomorphism f : B/A → D/C.
E. 3 Restriction to Subcategories 49 are commutative. If v : d → e is a morphism in RD, the image of the sequence (ρd )d ∈S by the map r D D r r IndD D (F )(v) : IndD (F )(d) → IndD (F )(e) is the sequence (σd )d ∈S deﬁned by σd = ρd ◦ HomRD (v, d ) . In other words σd is the map from HomRD (e, d ) to F (d ) deﬁned by ∀u ∈ HomRD (e, d ), σd (u) = ρd (u ◦ v) . D Clearly, the correspondences F → l IndD (F ) and F → r IndD D (F ) are Rlinear functors from FD ,R to FD,R . 3. Proposition : Let D ⊆ D be subcategories of C, both containing group isomorphisms, and let R be a commutative ring.