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Frobenius categories versus Brauer blocks: the Grothendieck by Lluís Puig

By Lluís Puig

This ebook contributes to special questions in glossy illustration thought of finite teams. at the one hand, it introduces and develops the summary environment of the Frobenius different types (also referred to as the Saturated fusion structures within the literature), created by way of the writer fifteen years in the past for a greater realizing of what was once loosely known as the neighborhood thought both of finite teams or of blocks, and for the aim of an eventual classification.

On the opposite hand, it offers the applying of the summary atmosphere to the blocks. specifically, it develops a framework for a deeper realizing of 1 of the relevant open difficulties in illustration concept, often called Alperin’s Weight Conjecture (AWC). one of many major result of the booklet is a discount theorem of the author’s personal type of AWC to quasi-simple groups.

Although it's a study monograph instead of a textbook, all of the arguments are broadly built to make it available to graduate scholars. a protracted advent provides a motivating perception to every bankruptcy and offers a easy guideline.

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Additional info for Frobenius categories versus Brauer blocks: the Grothendieck group of the Frobenius category of a Brauer block

Example text

1. 31 Our general notation mainly concerns group theory — our standard reference being [28] — and homological algebra — our standard reference being [18]. In particular, if G is a finite group, recall that Op (G) , Op (G) , Op (G) and Op (G) respectively denote the minimal or the maximal normal subgroups of G with their index or their order being a power of p or prime ˆ except to p ; note that this notation still makes sense for a finite k ∗ -group G ˆ that Op (G) remains a p-group. For any pair of subgroups H and K of G , we denote by TG (K, H) the set of x ∈ G fulfilling xKx−1 ⊂ H .

But, as a matter of fact, when dealing with contravariant functors a from F to Ab (cf.

3), they coincide (cf. 4). 7 F(b,G) is a Frobenius P -category. 1; thus, since p does not divide |NG (P, e)/P ·CG (P )| (cf. 1. Let Q be a subgroup of P , K a subgroup of Aut(Q) containing FQ (Q) and ϕ : Q → P an F(b,G) -morphism such that ϕ(Q) is fully ϕK-normalized in F(b,G) ; in particular, denoting by f the block of CG (Q) such that (P, e) contains (Q, f ) , there is x ∈ G fulfilling (cf. 2. As above, denote by g the block of CG NPK (Q) such that (P, e) contains the Brauer (b, G)-pair (NPK (Q), g) .

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