Jordan's theorem on finite linear groups

Jordan's theorem on a linear linear group theorem guarantees the existence of a large commutative subgroup in any finite linear group .

In its original form, proved by Camille Jordan , later improved several times.

Wording

For any dimension $n$ ${\ displaystyle n}$ , there is a number $f(n)$ ${\ displaystyle f (n)}$ such that any finite subgroup $G$ ${\ displaystyle G}$ groups $\mathrm {GL} (n,\mathbb {C} )$ ${\ displaystyle \ mathrm {GL} (n, \ mathbb {C})}$ invertible matrices with complex components contains a normal commutative subgroup $H$ ${\ displaystyle H}$ with index $[G:H]\leq f(n)$ ${\ displaystyle [G: H] \ leq f (n)}$

Variations and generalizations

Schur proved a more general result for periodic groups , while
$f(n)=\left({\sqrt {8n}}+1\right)^{2n^{2}}-\left({\sqrt {8n}}-1\right)^{2n^{2}}$ {\ displaystyle f (n) = \ left ({\ sqrt {8n}} + 1 \ right) ^ {2n ^ {2}} - \ left ({\ sqrt {8n}} - 1 \ right) ^ {2n ^ {2}}}

For finite groups, proved a more accurate estimate:
$f(n)=n!\cdot 12^{n(\pi (n+1)+1)}$ ${\ displaystyle f (n) = n! \ cdot 12 ^ {n (\ pi (n + 1) +1)}}$

Where

\pi (n)

{\ displaystyle \ pi (n)}

there is a distribution function of primes .

This rating was improved by , who replaced “12” with “6”.
Subsequently, Michael Collins, using the classification of finite simple groups , showed that $f(n)=(n+1)!$ ${\ displaystyle f (n) = (n + 1)!}$ at $n\geq 71$ ${\ displaystyle n \ geq 71}$ , and gave almost complete descriptions of behavior $f(n)$ ${\ displaystyle f (n)}$ at small $n$ ${\ displaystyle n}$ .

Jordan's theorem on finite linear groups

Wording

Variations and generalizations

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