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

Camille Jordan

Marie Ennemond Camille Jordan was a French mathematician, known both for his foundational work in group theory and for his influential Cours d'analyse. He was born in Lyon and educated at the École polytechnique...

. In that form, it states that there is a function ƒ(n) such that given a finite group

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

 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

Abelian group

In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...

, H is normal

Normal subgroup

In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....

 with respect to G and H has index

Index of a subgroup

In mathematics, specifically group theory, the index of a subgroup H in a group G is the "relative size" of H in G: equivalently, the number of "copies" of H that fill up G. For example, if H has index 2 in G, then intuitively "half" of the elements of G lie in H...

 at most ƒ(n). Schur

Issai Schur

Issai Schur was a mathematician who worked in Germany for most of his life. He studied at Berlin...

 proved a more general result that applies when G is assumed not to be finite but just periodic

Periodic group

In group theory, a periodic group or a torsion group is a group in which each element has finite order. All finite groups are periodic. The concept of a periodic group should not be confused with that of a cyclic group, although all finite cyclic groups are periodic.The exponent of a periodic group...

. Schur showed that ƒ(n) may be taken to be1/2 + 1)²ⁿ² − ((8n)1/2 − 1)²ⁿ².

A tighter bound (for n ≥ 3) is due to Speiser

Andreas Speiser

Andreas Speiser was a Swiss Mathematician and Philosopher of Science.-Life and work:Speiser studied since 1904 in Göttingen notably with David Hilbert, Felix Klein, Hermann Minkowski. In 1917 he became full time professor at the University of Zurich but later relocated in Basel...

 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 Blitchfeldt who replaced the "12" with a "6". Unpublished work on the finite case was also done by Boris Weisfeiler

Boris Weisfeiler

Boris Weisfeiler is a Russian-born mathematician who lived in the United States before going missing in Chile in 1985, aged 43. The Chilean government claimed that he drowned, but his family believes he was forced to disappear near Colonia Dignidad, an enclave led by ex-Nazi Paul Schäfer.-...

. Subsequently, Michael Collins using the classification of finite simple groups

Classification of finite simple groups

In mathematics, the classification of the finite simple groups is a theorem stating that every finite simple group belongs to one of four categories described below. These groups can be seen as the basic building blocks of all finite groups, in much the same way as the prime numbers are the basic...

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.

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