# Jordan–Schur theorem

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 group G that is a subgroup of the group of n-by-n complex matrices, then there is a subgroup H of G such that H is abelian, H is normal with respect to G and H has index at most ƒ(n). Schur proved a more general result that applies when G is assumed not 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.[1]

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.[1][2] 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 Unpublished work on the finite case was also done by Boris Weisfeiler.[3] Subsequently, Michael Collins using the classification of finite simple groups showed that in the finite case, one can take f(n)) = (n+1)! when n is at least 71, and gave near complete descriptions of the behavior for smaller n.

## References

1. ^ a b Curtis, Charles; Reiner, Irving (1962). Representation Theory of Finite Groups and Associated Algebras. John Wiley & Sons. pp. 258–262.
2. ^ 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.
3. ^ 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.
Notes