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! 12^n(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 Unpublished work on the finite case was also done by Boris Weisfeiler.^[3] Subsequently, Michael Collins, using the classification of finite simple groups, showed that in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.

See also

• Burnside's problem