Brauer's height zero conjecture

Let $G$ be a finite group and $p$ a prime. The set $\{\backslash rm\; Irr\}(G)$ of irreducible complex characters can be partitioned into Brauer $p$-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 $\backslash mathrm\{Irr\}(B)$.

Let $\backslash nu$ be the discrete valuation defined on the integers by $\backslash nu(mp^a)=a$ where $m$ is coprime to $p$. Brauer proved that if $B$ is a block with defect group $D$ then $\backslash nu(\backslash chi(1))\backslash geq\; \backslash nu(|G:D|)$ for each $\backslash chi\backslash in\backslash mathrm\{Irr\}(B)$. Brauer's Height Zero Conjecture asserts that $\backslash nu(\backslash chi(1))=\backslash nu(|G:D|)$ for all $\backslash chi\backslash in\backslash mathrm\{Irr\}(B)$ if and only if $D$ is abelian.

Brauer's Height Zero Conjecture was formulated by Richard Brauer in 1955. {{cite book| last1 = Brauer | first1 = Richard D. | year = 1956 | title = Proceedings of the International Symposium on Algebraic Number Theory, Tokyo and Nikko, 1955 | chapter = Number theoretical investigations on groups of finite order | publisher=Science Council of Japan| pages=55–62}} It also appeared as Problem 23 in Brauer's list of problems.{{cite book | last1 = Brauer | first1 = Richard D. | year = 1963 | chapter = Representations of finite groups|title= Lectures in Mathematics

| volume=1 | publisher= Wiley | pages=133-175 }} Brauer's Problem 12 of the same list asks whether the character table of a finite group $G$ determines if its Sylow $p$-subgroups are abelian. Solving Brauer's height zero conjecture for blocks whose defect groups are Sylow $p$-subgroups (or equivalently, that contain a character of degree coprime to $p$) 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{{cite journal |last1=Kessar|first1=Radha|last2=Malle|first2=Gunter|title=Quasi-isolated blocks and Brauer's height zero conjecture|doi=10.4007/annals.2013.178.1.6 |journal=Annals of Mathematics |volume=178 |date=2013|pages=321-384|arxiv=1112.2642}} in 2013 after a reduction to finite simple groups by Thomas R. Berger and Reinhard Knörr.{{cite journal | last1=Berger | first1=Thomas R. | last2=Knörr | first2=Reinhard | title=On Brauer's height 0 conjecture | journal=Nagoya Mathematical Journal | volume=109 | date=1988 | pages=109–116 | doi=10.1017/S0027763000002798 }}

The only if direction was proved for $p$-solvable groups by David Gluck and Thomas R. Wolf.{{cite journal|last1=Gluck|first1=David|last2=Wolf|first2=Thomas R.|title=Brauer's height conjecture for p-solvable groups

|journal=Transactions of the American Mathematical Society|volume=282|date=1984|pages=137–152|doi=10.2307/1999582

}} 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.{{cite journal | last1 = Navarro | first1 = Gabriel | last2 = Tiep | first2 = Pham Huu | title = Characters of relative $p\text{'}$-degree over normal subgroups | journal = Annals of Mathematics | volume = 178 | year = 2013 | pages = 1135–1171 | doi = 10.4007/annals.2013.178}} Gabriel Navarro and Britta Späth showed that the so-called inductive Alperin—McKay condition for simple groups implied Brauer's Height Zero Conjecture.{{cite journal|last1=Navarro|first1=Gabriel| last2=Späth|first2=Britta|

title=On Brauer's height zero conjecture|

journal=Journal of the European Mathematical Society|volume=16|date=2014|pages=695–747|doi=10.4171/JEMS/444|arxiv=2209.04736}} Lucas Ruhstorfer completed the proof of these conditions for the case $p=2$.{{cite journal|last=Ruhstorfer|first=Lucas|date=2022|title=The Alperin-McKay conjecture for the prime 2|journal=Annals of Mathematics| volume = 201(2) | pages = 379–457 | arxiv = 2204.06373}} 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.{{cite journal|author=Malle, Gunter|author2=Navarro, Gabriel|author3= Schaeffer Fry, A. A.|author4=Tiep, Pham Huu|title=Brauer's Height Zero Conjecture|year=2024|doi=10.4007/annals.2024.200.2.4 |volume=200 |journal=Annals of Mathematics |pages=557-608|arxiv=2209.04736}}

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Category:Conjectures that have been proved