View source for Brauer height-zero conjecture

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

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120450/b1204501.png" /> be an irreducible character in a block <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120450/b1204502.png" /> of a [[Group|group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120450/b1204503.png" /> with defect group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120450/b1204504.png" /> (cf. also [[Defect group of a block|Defect group of a block]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120450/b1204505.png" /> be the discrete valuation defined on the integers with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120450/b1204506.png" /> whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120450/b1204507.png" /> is prime to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120450/b1204508.png" />. By a theorem of Brauer, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120450/b1204509.png" />. The height of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120450/b12045010.png" /> is defined to be

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120450/b12045011.png" /></td> </tr></table>

Every block contains an irreducible character of height zero. Brauer's height-zero conjecture is the assertion that every irreducible character in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120450/b12045012.png" /> has height zero if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120450/b12045013.png" /> is Abelian (cf. also [[Abelian group|Abelian group]]).

The conjecture is still (1998) open. It has been proven for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120450/b12045014.png" />-solvable groups (cf. also [[Pi-solvable group|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120450/b12045015.png" />-solvable group]]) by combined work of P. Fong (see [[#References|[a2]]], X.4), and D. Gluck and T. Wolf [[#References|[a3]]]. The  "if"  direction has been reduced to the consideration of quasi-simple groups by T.R. Berger and R. Knörr [[#References|[a1]]]. It has been checked for some of these, but not all. The evidence for the  "only if"  direction is more slender.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T.R. Berger,   R. Knörr,   "On Brauer's height <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120450/b12045016.png" /> conjecture"  ''Nagoya Math. J.'' , '''109'''  (1988)  pp. 109–116</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Feit,   "The representation theory of finite groups" , North-Holland  (1982)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  D. Gluck,   T.R. Wolf,   "Brauer's height conjecture for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120450/b12045017.png" />-solvable groups"  ''Trans. Amer. Math. Soc.'' , '''282''' :  1  (1984)  pp. 137–152</TD></TR></table>