By serge Bouc

ISBN-10: 364211296X

ISBN-13: 9783642112966

ISBN-10: 3642112978

ISBN-13: 9783642112973

This quantity exposes the idea of biset functors for finite teams, which yields a unified framework for operations of induction, restrict, inflation, deflation and delivery through 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 concerns 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 comprises the constitution of the functor of rational representations and rational p-biset functors. The final chapters reveal 3 purposes of biset functors to long-standing open difficulties, specifically the constitution of the Dade workforce of an arbitrary finite p-group.This ebook is meant either to scholars and researchers, because it offers a didactic exposition of the fundamentals and a rewriting of complex leads to the realm, with a few new principles and proofs.

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This quantity exposes the speculation of biset functors for finite teams, which yields a unified framework for operations of induction, limit, inflation, deflation and delivery by means of isomorphism. the 1st half remembers the fundamentals on biset different types and biset functors. the second one half is anxious with the Burnside functor and the functor of complicated characters, including semisimplicity concerns and an summary of eco-friendly biset functors.

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**Extra info for Biset functors for finite groups**

**Example text**

D If F is a simple object of FD,R , and if ResD D F = {0}, then ResD F is a simple object of FD ,R . Proof: Let M be a non zero subfunctor of ResD D F . By adjunction, the (non zero) inclusion morphism M → ResD F yields a non zero morphism D D s : l IndD M → F . The image of s is a non zero subfunctor of F , so it is equal to F , and s is D surjective. Hence ResD D s is surjective, since the functor ResD is exact. So the map D D l D ResD D s : ResD ◦ IndD M → ResD F D l is surjective. 4, the functor ResD D ◦ IndD is isomorD phic to the identity functor.

2. If Y is a G-set, then the map βU,Y : U ×G Y → U ×G Y ×G U op given by βU,Y : h, (u,G y) → hu,G (1, y),G u , for h ∈ H, u ∈ U , and y ∈ Y , is a well deﬁned morphism of (H, H)bisets. 3. If U is left-free, then αU,X is injective, for any X, and βU,Y is injective, for any Y . 4. If U is left-transitive, then αU,X is surjective, for any X, and βU,Y is surjective, for any Y . Proof: If u ∈ U , denote by uop the element u, viewed in the (G, H)-biset U op . With this notation, if g ∈ G and h ∈ H, then (hug)op = g −1 uop h−1 .

10). 2. 2 holds in D. 3. Deﬁnition : A subcategory D of C is called admissible if it contains group isomorphisms, and if the following conditions are fulﬁlled: A1. If G and H are objects of D, then there is a subset S(H, G) of the set of subgroups of H × G, invariant under (H × G)-conjugation, such that HomD (G, H) is the subgroup of HomC (G, H) generated by the elements [(H × G)/L], for L ∈ S(H, G). A2. If G and H are objects of D, and if L ∈ S(H, G), then q(L) is an object H of D. Moreover DefresG p2 (L)/k2 (L) and Indinf p1 (L)/k1 (L) are morphisms in D, in other words { xk2 (L), x | x ∈ p2 (L)} ∈ S p2 (L)/k2 (L), G { x, xk1 (L) | x ∈ p1 (L)} ∈ S H, p1 (L)/k1 (L) .

### Biset functors for finite groups by serge Bouc

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