# Brauer height-zero conjecture

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2010 Mathematics Subject Classification: Primary: 20C20 Secondary: 20C33 [MSN][ZBL]

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

Let $\chi$ be an irreducible character in a block $B$ of a group $G$ with defect group $D$ (cf. also Defect group of a block). Let $\nu$ be the discrete valuation defined on the integers with $\def\a{\alpha}\nu(np^\a)=\a$ whenever $n$ is prime to $p$. By a theorem of Brauer, $\nu(\chi(1)\ge \nu(|G:D|)$. The height of $\chi$ is defined to be

$$\nu(\chi(1))-\nu(|G:D|).$$ Every block contains an irreducible character of height zero. Brauer's height-zero conjecture is the assertion that every irreducible character in $B$ has height zero if and only if $D$ is Abelian (cf. also Abelian group).

That every irreducible character in $B$ has height zero when $D$ is Abelian was proved for $p$-solvable groups (cf. also $\pi$-solvable group) by P. Fong (see [Fe], X.4). The converse for $p$-solvable groups was proved by D. Gluck and T. Wolf [GlWo], 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 [BeKn]. The task of checking this half of the conjecture for the quasisimple groups was completed in 2011 by R. Kessar and G. Malle [KeMa], hence completing the proof of the "if" direction. The evidence for the "only if" direction is more slender.