Titlebook: Blocks of Finite Groups and Their Invariants; Benjamin Sambale Book 2014 Springer International Publishing Switzerland 2014 20C15,20C20,20

以烟熏消毒

Definitions and Facts of groups algebras, (lower) defect groups, the Brauer homomorphism, decomposition numbers, subsections, and fusion systems. Moreover, we present Brauer’s three main theorems as well as a few other important results. Most theorems are given without proof.

transplantation

assent

Ovulation

Metacyclic Defect Groupslt due to various authors. In the odd case we give a proof of Brauer’s .(.)-Conjecture, Olsson’s Conjecture and Brauer’s Height Zero Conjecture. Moreover, we use a recent result by Watanabe to describe blocks with metacyclic, minimal non-abelian defect groups.

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Bicyclic Groupsups. In this chapter we classify all saturated fusion systems on bicyclic groups. For odd primes, every bicyclic group is metacyclic and the classification is due to Stancu. In case . = 2 the classification is very delicate. As an application we verify Olsson’s Conjecture for blocks with bicyclic defect groups.

彻底明白

genesis

Abelian Defect Groupsnd for blocks with abelian defect groups. The proof uses results about regular orbits under coprime actions. Moreover, we show that Brauer’s .(.)-Conjecture holds for blocks with abelian defect groups if the inertial index is less than 256.

ARBOR

Life in the Solar System and BeyondWe introduce a list of open conjectures about block theory of finite groups. The first of these conjectures was proposed in 1954 by Brauer, and the last one of our list is a conjecture by Gluck from 2011. It also includes famous conjectures by Olsson, Alperin, McKay and others. All of these conjectures will be considered in the following chapters.

BOOST

Dirk Schulze-Makuch,Louis N. IrwinWe introduce a quadratic form arising from the Cartan matrix of a block of a finite group. By invoking Brauer’s notion of basic sets, we exploit some properties of the quadratic form with will lead to restriction on the number of characters of the block. We also discuss a question about the indecomposability of Cartan matrices.