Quotients and normal subcategories in Frobenius P-categories

Part of the Progress in Mathematics book series (PM, volume 274)


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 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:PP′ is (F,F′)-functorial whenever, for any pair of subgroups Q and R of P and any ϕ ∈ F(Q,R), we have
$$ \phi (R \cap K(\alpha )) \subset Ker (\alpha )$$
and the group homomorphism ϕ′ :α(R) → α(Q) determined by ϕ belongs to F′(α(Q), α(R)). In this case, α determines an evident functor
$$ \mathfrak{f}_\alpha :F \to F'$$
that we call 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 fσ.


Normal Subgroup Small Element Group Homomorphism Proper Subgroup Chevalley Group 
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© Birkhäuser Verlag AG 2009

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