# 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 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:

*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 assumed not to be finite, but just periodic. Schur showed that *ƒ*(*n*) may be taken to be

- ((8
*n*)^{1/2}+ 1)^{2n2}− ((8*n*)^{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*!12^{n(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 Unpublished work on the finite case was also done by Boris Weisfeiler.[3] Subsequently, Michael Collins, using the classification of finite simple groups, showed that in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 *ƒ*(*n*) = (*n*+1)! when *n* is at least 71, and gave near complete descriptions of the behavior for smaller *n*.

## See also

## References

- 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.