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 GL(n, C) of invertible n-by-n complex matrices, there is a subgroup H of G with the following properties:
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. This was subsequently improved by Blichfeldt who replaced the "12" with a "6". Unpublished work on the finite case was also done by Boris Weisfeiler. 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.
- Curtis, Charles; Reiner, Irving (1962). Representation Theory of Finite Groups and Associative Algebras. John Wiley & Sons. pp. 258–262.
- Speiser, Andreas (1945). Die Theorie der Gruppen von endlicher Ordnung, mit Andwendungen auf algebraische Zahlen und Gleichungen sowie auf die Krystallographie, von Andreas Speiser. New York: Dover Publications. pp. 216–220.
- Collins, Michael J. (2007). "On Jordan's theorem for complex linear groups". Journal of Group Theory. 10 (4): 411–423. doi:10.1515/JGT.2007.032.