View source for Brauer's k(B) conjecture

{{Short description|Conjecture in modular representation theory}}

'''[[Richard Brauer]]'s k(B) Conjecture''' is a conjecture in [[modular representation theory]] of finite groups relating the number of complex irreducible [[Character theory|characters]] in a [[Modular representation theory|Brauer block]] and the order of its defect groups. It was first announced in 1955.<ref> {{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 |oclc=39212542 }}</ref> It is Problem 20 in Brauer's list of problems.<ref>{{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 |mr=0178056 |oclc=523576 }}</ref>
==Statement==
Let <math>G</math> be a finite group and <math>p</math> a prime. The set <math>{\rm Irr}(G)</math> of irreducible complex [[Character theory|characters]] can be partitioned into <math>p</math>-[[Modular representation theory#Blocks and the structure of the group algebra|blocks]]. To each <math>p</math>-block <math>B</math> is canonically associated a conjugacy class of [[p-group|<math>p</math>-subgroups]], called the [[Modular representation theory#Defect groups|defect groups]] of <math>B</math>. The set of irreducible characters belonging to <math>B</math> is denoted by <math>\mathrm{Irr}(B)</math>.

The '''k(B) Conjecture''' asserts that

<math>|{\rm Irr}(B)|\leq|D|</math>.

==The k(GV) problem==

In the case of blocks of <math>p</math>-[[Solvable groups#p-solvable|solvable groups]], the conjecture is equivalent to the following question.<ref>{{cite journal |last1=Nagao |first1=Hirosi |title=On a conjecture of Brauer for p-solvable groups |journal=Journal of Mathematics |volume=13 |issue=1 |date=1962 |pages=35–38 |mr=0152569 |url=https://projecteuclid.org/journals/osaka-journal-of-mathematics/volume-13/issue-1/On-a-conjecture-of-Brauer-for-p-solvable-groups/ojm/1353055010.full }}</ref> Let <math>V</math> be an [[elementary abelian group]] of order <math>p^d</math>, let <math>G</math> be a finite group of order non-divisible by <math>p</math> and [[Group action#Notable_properties_of_actions|acting faithfully]] on <math>V</math> by group automorphisms. Let <math>GV</math> denote the associated semidirect product and let <math>k(GV)</math> be its number of conjugacy classes. Then

<math>k(GV)\leq |V|.</math>

This was proved by [[John G. Thompson|John Thompson]] and Geoffrey Robinson,<ref>{{cite journal |last1=Robinson |first1=Geoffrey R. |last2=Thompson |first2=John G. |title=On Brauer'sk(B)-Problem |journal=Journal of Algebra |date=September 1996 |volume=184 |issue=3 |pages=1143–1160 |doi=10.1006/jabr.1996.0304 }}</ref> except for finitely many prime numbers. A proof of the last open cases was published in 2004<ref>{{cite journal |last1=Gluck |first1=David |last2=Magaard |first2=Kay |last3=Riese |first3=Udo |last4=Schmid |first4=Peter |title=The solution of the k(GV)-problem |journal=Journal of Algebra |date=September 2004 |volume=279 |issue=2 |pages=694–719 |doi=10.1016/j.jalgebra.2004.02.027 }}</ref><ref>{{cite book |doi=10.1142/9781860949715 |title=The Solution of the k(GV) Problem |series=ICP Advanced Texts in Mathematics |date=2007 |volume=4 |isbn=978-1-86094-971-5 |first1=Peter |last1=Schmid }}</ref>

==References==
{{reflist}}

[[Category:Conjectures that have been proved]]
[[Category:Module theory]]
[[Category:Representation theory of finite groups]]