Jordan-Schur's theorem

The set of Jordan Schur , also known as the set of Jordan on finite linear groups known, a mathematical theorem , which in its original form by Camille Jordan comes. In this form it says that there is a function such that for every finite subgroup of the general linear group there is a subgroup such that: ${\displaystyle f\colon \mathbb {N} \rightarrow \mathbb {N} }$ ${\displaystyle G}$ ${\displaystyle \mathrm {GL} (n,\mathbb {C} )}$ ${\displaystyle H\subset G}$

• ${\displaystyle H}$is abelian ,
• ${\displaystyle H}$is a normal divisor of , ${\displaystyle G}$
• the index of in is met . ${\displaystyle H}$ ${\displaystyle G}$ ${\displaystyle [G:H]\leq f(n)}$

Issai Schur had achieved a more general result by no longer presupposing the finiteness of the group, but only that it is a torsion group . Schur showed that you can

${\displaystyle f(n)=((8n)^{\frac {1}{2}}+1)^{2n^{2}}-((8n)^{\frac {1}{2}}-1)^{2n^{2}}}$

can take. Speiser received the better estimate for and under the condition of finiteness ${\displaystyle n\geq 3}$ ${\displaystyle G}$

${\displaystyle f(n)=n!\cdot 12^{n(\pi (n+1)+1)}}$,

where is the prime function . In a further improvement, Blichfeldt was able to replace 12 with 6 in the above formula. Finally, assuming the finiteness of finite simple groups using the classification theorem , Michael Collins showed that one can take for the estimating function , and gave an almost complete description of the behavior for small ones . ${\displaystyle \pi }$ ${\displaystyle G}$ ${\displaystyle n\geq 71}$ ${\displaystyle f(n)=(n+1)!}$ ${\displaystyle n}$

Individual evidence

1. ↑ Charles W. Curtis , Irving Reiner : Representation Theory of Finite Groups and Associative Algebras , AMS Chelsea Publishing (1962), ISBN 0-8218-4066-5 , Theorem (36,14).
2. ^ Charles W. Curtis , Irving Reiner : Representation Theory of Finite Groups and Associative Algebras , AMS Chelsea Publishing (1962), ISBN 0-8218-4066-5 , end of Chapter V.36: Theorems of Jordan, Burnside, and Schur on Linear Groups.
3. ↑ A. Speiser: The theory of groups of finite order, with applications to algebraic numbers and equations and to crystallography , New York: Dover Publications (1945), pp. 216-220.
4. ^ MJ Collins: On Jordan's theorem for complex linear groups , Journal of Group Theory (2007), Volume 10 (4), pages 411-423.