Brauer's k(B) conjecture

Let ${\displaystyle G}$ be a finite group and ${\displaystyle p}$ a prime. The set ${\displaystyle {\rm {Irr}}(G)}$ of irreducible complex characters can be partitioned into ${\displaystyle p}$-blocks. To each ${\displaystyle p}$-block ${\displaystyle B}$ is canonically associated a conjugacy class of ${\displaystyle p}$-subgroups, called the defect groups of ${\displaystyle B}$. The set of irreducible characters belonging to ${\displaystyle B}$ is denoted by ${\displaystyle \mathrm {Irr} (B)}$.

The k(B) Conjecture asserts that

${\displaystyle |{\rm {Irr}}(B)|\leq |D|}$.

In the case of blocks of ${\displaystyle p}$-solvable groups, the conjecture is equivalent to the following question.^[3] Let ${\displaystyle V}$ be an elementary abelian group of order ${\displaystyle p^{d}}$, let ${\displaystyle G}$ be a finite group of order non-divisible by ${\displaystyle p}$ and acting faithfully on ${\displaystyle V}$ by group automorphisms. Let ${\displaystyle GV}$ denote the associated semidirect product and let ${\displaystyle k(GV)}$ be its number of conjugacy classes. Then

${\displaystyle k(GV)\leq |V|.}$

This was proved by John Thompson and Geoffrey Robinson,^[4] except for finitely many prime numbers. A proof of the last open cases was published in 2004.^[5][6]