Brauer's k(B) conjecture explained

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 1955.^[1] It is Problem 20 in Brauer's list of problems.^[2]

Statement

Let

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

The k(B) Conjecture asserts that

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

The k(GV) problem

In the case of blocks of

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

Notes and References

1. Book: Brauer . Richard D. . 1956 . Proceedings of the International Symposium on Algebraic Number Theory, Tokyo and Nikko, 1955 . Number theoretical investigations on groups of finite order . Science Council of Japan. 55–62 . 39212542 .
2. Book: Brauer . Richard D. . 1963 . Representations of finite groups . Lectures in Mathematics . 1 . Wiley . 133–175 . 0178056 . 523576 .
3. Nagao . Hirosi . On a conjecture of Brauer for p-solvable groups . Journal of Mathematics . 13 . 1 . 1962 . 35–38 . 0152569 .
4. Robinson . Geoffrey R. . Thompson . John G. . On Brauer'sk(B)-Problem . Journal of Algebra . September 1996 . 184 . 3 . 1143–1160 . 10.1006/jabr.1996.0304 .
5. Gluck . David . Magaard . Kay . Riese . Udo . Schmid . Peter . The solution of the k(GV)-problem . Journal of Algebra . September 2004 . 279 . 2 . 694–719 . 10.1016/j.jalgebra.2004.02.027 .
6. Book: 10.1142/9781860949715 . The Solution of the k(GV) Problem . ICP Advanced Texts in Mathematics . 2007 . 4 . 978-1-86094-971-5 . Peter . Schmid .