Richard Brauer's k(B) Conjecture is a conjecture in modular representation theory of finite groups relating the number of complex irreducible characters in a Brauer block and the order of its defect groups. It was first announced in 1946. It is Problem 20 in Brauer's list of problems.
Statement Let G {\displaystyle G} be a finite group and p {\displaystyle p} a prime. The set I r r ( G ) {\displaystyle {\rm {Irr}}(G)} of irreducible complex characters can be partitioned into p {\displaystyle p} -blocks. To each p {\displaystyle p} -block B {\displaystyle B} is canonically associated a conjugacy class of p {\displaystyle p} -subgroups, called the defect groups of B {\displaystyle B} . The set of irreducible characters belonging to B {\displaystyle B} is denoted by I r r ( B ) {\displaystyle \mathrm {Irr} (B)} . The k(B) Conjecture asserts that
| I r r ( B ) | ≤ | D | {\displaystyle |{\rm {Irr}}(B)|\leq |D|} .
The k(GV) problem In the case of blocks of p {\displaystyle p} -solvable groups, the conjecture is equivalent to the following question. Let V {\displaystyle V} be an elementary abelian group of order p d {\displaystyle p^{d}} , let G {\displaystyle G} be a finite group of order non-divisible by p {\displaystyle p} and acting faithfully on V {\displaystyle V} by group automorphisms. Let G V {\displaystyle GV} denote the associated semidirect product and let k ( G V ) {\displaystyle k(GV)} be its number of conjugacy classes. Then
k ( G V ) ≤ | V | . {\displaystyle k(GV)\leq |V|.}
This was proved by John Thompson and Geoffrey Robinson, except for finitely many prime numbers. A proof of the last open cases was published in 2004.
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