Biset Functors for Finite Groups (Lecture Notes in by serge Bouc

By serge Bouc

This quantity exposes the idea of biset functors for finite teams, which yields a unified framework for operations of induction, restrict, 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 top quantity p. This contains the constitution of the functor of rational representations and rational p-biset functors. The final chapters disclose 3 functions of biset functors to long-standing open difficulties, specifically 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 complicated ends up in the realm, with a few new rules and proofs.

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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.

This shows the existence of the first exact sequence in Assertion 1. Now an element k of K is in k1 (M ∗ L) if and only if there exists h ∈ H such that (k, h) ∈ M and (h, 1) ∈ L. Thus h ∈ p2 (M ) ∩ k1 (L). This shows that the image of the group k1 (M ∗ L) × 1 by the morphism θ is precisely equal to p2 (M ) ∩ k1 (L) / k2 (M ) ∩ k1 (L) . Moreover its intersection with the kernel k1 (M ) × k2 (L) of θ is equal to k1 (M ) × 1. This yields the second exact sequence of Assertion 1. The third one is similar.

11. Remark : If D contains group isomorphisms, then in particular the equivalence classes of objects of D modulo group isomorphism form a subset of the set of isomorphism classes of finite groups. 12. Corollary : If D contains group isomorphisms, then the category FD,R has enough projective objects. 46 3 Biset Functors Proof: For any object D of D, the Yoneda functor YD : G → HomRD (D, G) is a projective object of FD,R : indeed, evaluation at D preserves exact sequences, and by Yoneda’s lemma, for any object F of FD,R , the map f ∈ F (D) → φD,f ∈ HomFD,R (YD , F ) is an isomorphism, where the evaluation of φD,f at an object G of D is defined by φD,f,G : u ∈ YD (G) = HomRD (D, G) → F (u)(f ) ∈ F (G) .

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