Frobenius Categories versus Brauer Blocks pp 179-193 | Cite as

# Quotients and normal subcategories in Frobenius *P*-categories

Chapter

## Abstract

12.1 Let us come back to our abstract setting. Before going further into the blocks, we need to develop the analogy between our abstract setting and the groups by discussing
and the group homomorphism ϕ′ :α(
that we call

*homomorphisms, quotients*and*normal*structures. Let*P*be a finite*p*-group and F a*P*-category; if*P*′ is a second finite*p*-group and F′ a*P*′-category, we say that a group homomorphism α:*P*→*P*′ is (F,F′)-*functorial*whenever, for any pair of subgroups*Q*and*R*of*P*and any ϕ ∈ F(*Q,R*), we have$$
\phi (R \cap K(\alpha )) \subset Ker (\alpha )$$

(12.1.1)

*R*) → α(*Q*) determined by ϕ belongs to F′(α(*Q*), α(*R*)). In this case, α determines an evident functor$$
\mathfrak{f}_\alpha :F \to F'$$

(12.1.2)

*Frobenius functor*; clearly, the composition of Frobenius functors is a Frobenius functor. If*P*′ =*P*and F′=F then id_{ P }is obviously (F, F)-functorial and, if F is divisible, any σ ∈ F(*P*) is (F, F)-functorial and determines an evident*natural*isomorphism id_{ F }≌f_{σ}.## Keywords

Normal Subgroup Small Element Group Homomorphism Proper Subgroup Chevalley Group
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Birkhäuser Verlag AG 2009