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 group G that is a subgroup of the group of n-by-n complex matrices, then there is a subgroup H of G such that H is abelian, H is normal with respect to G and H has index at most 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

((8n)^1/2 + 1)²ⁿ² − ((8n)^1/2 − 1)²ⁿ².

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. 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 Unpublished work on the finite case was also done by Boris Weisfeiler. Subsequently, Michael Collins using the classification of finite simple groups showed that in the finite case, one can take f(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