The Brauer Height Zero Conjecture is a conjecture in modular representation theory of finite groups relating the degrees of the complex irreducible characters in a Brauer block and the structure of its defect groups. It was formulated by Richard Brauer in 1955.
Let be a finite group and a prime. The set of irreducible complex characters can be partitioned into Brauer -blocks. To each -block is canonically associated a conjugacy class of -subgroups, called the defect groups of . The set of irreducible characters belonging to is denoted by .
Let be the discrete valuation defined on the integers by where is coprime to . Brauer proved that if is a block with defect group then for each . Brauer's Height Zero Conjecture asserts that for all if and only if is abelian.
Brauer's Height Zero Conjecture was formulated by Richard Brauer in 1955. [1] It also appeared as Problem 23 in Brauer's list of problems. [2] Brauer's Problem 12 of the same list asks whether the character table of a finite group determines if its Sylow -subgroups are abelian. Solving Brauer's height zero conjecture for blocks whose defect groups are Sylow -subgroups (or equivalently, that contain a character of degree coprime to ) also gives a solution to Brauer's Problem 12.
The proof of the if direction of the conjecture was completed by Radha Kessar and Gunter Malle [3] in 2013 after a reduction to finite simple groups by Thomas R. Berger and Reinhard Knörr. [4]
The only if direction was proved for -solvable groups by David Gluck and Thomas R. Wolf. [5] The so-called generalized Gluck—Wolf theorem, which was a main obstacle towards a proof of the Height Zero Conjecture was proven by Gabriel Navarro and Pham Huu Tiep in 2013. [6] Gabriel Navarro and Britta Späth showed that the so-called inductive Alperin—McKay condition for simple groups implied Brauer's Height Zero Conjecture. [7] Lucas Ruhstorfer completed the proof of these conditions for the case . [8] The case of odd primes was finally settled by Gunter Malle, Gabriel Navarro, A. A. Schaeffer Fry and Pham Huu Tiep using a different reduction theorem. [9]