Brauer height-zero conjecture

For notation and definitions, see also Brauer first main theorem.

Let be an irreducible character in a block of a group with defect group (cf. also Defect group of a block). Let be the discrete valuation defined on the integers with whenever is prime to . By a theorem of Brauer, . The height of is defined to be

Every block contains an irreducible character of height zero. Brauer's height-zero conjecture is the assertion that every irreducible character in has height zero if and only if is Abelian (cf. also Abelian group).

That every irreducible character in has height zero when is Abelian was proved for -solvable groups (cf. also -solvable group) by P. Fong (see [a2], X.4). The converse for -solvable groups was proved by D. Gluck and T. Wolf [a3], using the classification of finite simple groups. The "if" direction has been reduced to the consideration of quasi-simple groups by T.R. Berger and R. Knörr [a1]. The task of checking this half of the conjecture for the quasisimple groups was completed in 2011 by R. Kessar and G. Malle [a4], hence completing the proof of the "if" direction. The evidence for the "only if" direction is more slender.

References

[a1]	T.R. Berger, R. Knörr, "On Brauer's height conjecture" Nagoya Math. J. , 109 (1988) pp. 109–116
[a2]	W. Feit, "The representation theory of finite groups" , North-Holland (1982)
[a3]	D. Gluck, T.R. Wolf, "Brauer's height conjecture for -solvable groups" Trans. Amer. Math. Soc. , 282 : 1 (1984) pp. 137–152
[a4]	R. Kessar, G. Malle, "Quasi-isolated blocks and Brauer's height conjecture" arXiv:1112.2642