Jordan–Schur theorem
{{Short description|A mathematical theorem on finite linear groups}}
In mathematics, the Jordan–Schur theorem also known as Jordan's theorem on finite linear groups is a theorem in its original form due to Camille Jordan. In that form, it states that there is a function ƒ(n) such that given a finite subgroup G of the group {{nowrap|GL(n, C)}} of invertible n-by-n complex matrices, there is a subgroup H of G with the following properties:
- H is abelian.
- H is a normal subgroup of G.
- The index of H in G satisfies (G : H) ≤ ƒ(n).
Schur proved a more general result that applies when G is not assumed to be finite, but just periodic. Schur showed that ƒ(n) may be taken to be
:((8n)1/2 + 1)2n2 − ((8n)1/2 − 1)2n2.{{cite book |title=Representation Theory of Finite Groups and Associative Algebras |last1=Curtis |first1=Charles | author1-link = Charles W. Curtis |last2=Reiner|first2= Irving|author2-link=Irving Reiner |year=1962 |publisher=John Wiley & Sons |pages=258–262}}
A tighter bound (for n ≥ 3) is due to Speiser, who showed that as long as G is finite, one can take
:ƒ(n) = n! 12n(π(n+1)+1)
where π(n) is the prime-counting function.{{cite book|last=Speiser|first=Andreas|title=Die Theorie der Gruppen von endlicher Ordnung, mit Anwendungen auf algebraische Zahlen und Gleichungen sowie auf die Krystallographie, von Andreas Speiser|year=1945|publisher=Dover Publications|location=New York|pages=216–220}} This was subsequently improved by Hans Frederick Blichfeldt who replaced the 12 with a 6. Unpublished work on the finite case was also done by Boris Weisfeiler.{{cite journal|last=Collins|first=Michael J.|title=On Jordan's theorem for complex linear groups|journal=Journal of Group Theory|year=2007|volume=10|issue=4|pages=411–423|doi=10.1515/JGT.2007.032}} Subsequently, Michael Collins, using the classification of finite simple groups, showed that in the finite case, one can take ƒ(n) = (n + 1)! when n is at least 71, and gave near complete descriptions of the behavior for smaller n.