Characters and Blocks of Finite Groups

Abstract

This is a clear, accessible and up-to-date exposition of modular representation theory of finite groups from a character-theoretic viewpoint. After a short review of the necessary background material, the early chapters introduce Brauer characters and blocks and develop their basic properties. The next three chapters study and prove Brauer's first, second and third main theorems in turn. These results are then applied to prove a major application of finite groups, the Glauberman Z*-theorem. Later chapters examine Brauer characters in more detail. The relationship between blocks and normal subgroups is also explored and the modular characters and blocks in p-solvable groups are discussed. Finally, the character theory of groups with a Sylow p-subgroup of order p is studied. Each chapter concludes with a set of problems. The book is aimed at graduate students, with some previous knowledge of ordinary character theory, and researchers studying the representation theory of finite groups.

Irreducible characters of even degree and normal Sylow 2-subgroups

“…Since |P | is even and q is odd, by Maschke's theorem, X F q is completely reducible, and moreover, is faithful since X is faithful. Using the FongSwan theorem [23,Theorem 10.1] on lifts of irreducible Brauer characters in solvable groups, we conclude that P has a complex faithful character, say χ, of degree n. Now we apply [12, Theorem A] to deduce that the number of generators in a minimal generating set for P , say d(P ), is at most (3/2)(n − s) + s, where s is the number of linear constituents of χ. In particular, d(P ) ≤ [3n/2], and it follows that…”

A reduction theorem for the blockwise Alperin weight conjecture

“…For a given finite group G this defines IBr.G/, the set of irreducible p-Brauer characters of G, see [25,Chapter 2]. The quotient F D R=I is a field of characteristic p, and is the algebraic closure of its prime field F p .…”

“…For blocks and characters in general we use the notation as introduced in [13] and [25]. For a finite group G and any block b 2 Bl.G/ let b W Z.F G/ !…”

On p-Brauer characters of p′-degree and self-normalizing Sylow p-subgroups

“…Let T be the stabilizer of y in G. Since P c T and T is Q-invariant, by the Cli¤ord correspondence, we may easily assume that y is G-invariant. Now, by using [20,Theorem 8.28] for the triple ðGQ; N; yÞ, we can easily assume that N is a central p 0 -subgroup of G centralized by Q.…”