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 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.[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 Hans Frederick 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 Unpublished work on the finite case was also done by Boris Weisfeiler.[3] Subsequently, Michael Collins, using the classification of finite simple groups, showed that in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.
Generalization to arbitrary fields
[edit]The theorem has also been extended to finite subgroups of general linear groups over arbitrary fields - with a more complicated albeit analogous statement - by Collins[4] and separately Larsen and Pink.[5] Collins's paper includes exact bounds and depends on the classification of finite simple groups; Larsen and Pink used a different approach that avoids depending on the CFSG.
See also
[edit]References
[edit]- 1 2 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 Anwendungen 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.
- ↑ Collins, Michael J. (2008). "Modular analogues of Jordan's theorem for finite linear groups". J. Reine Angew. Math. 624: 143–171. arXiv:0709.3245. doi:10.1515/CRELLE.2008.084.
- ↑ Larsen, Michael J.; Pink, Richard (2011). "Finite subgroups of algebraic groups". J. Amer. Math. Soc. 24 (4): 1105–1158. doi:10.1090/S0894-0347-2011-00695-4.