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
Marie Ennemond Camille Jordan (; 5 January 1838 – 22 January 1922) was a French mathematician, known both for his foundational work in group theory and for his influential ''Cours d'analyse''.
Biography
Jordan was born in Lyon and educated at ...
. In that form, it states that there is a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
''ƒ''(''n'') such that given a finite
subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
''G'' of the
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
of
invertible
In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers.
Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...
''n''-by-''n''
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
matrices
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
, 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
:((8''n'')
1/2 + 1)
2''n''2 − ((8''n'')
1/2 − 1)
2''n''2.
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
Hans Frederick Blichfeldt who replaced the 12 with a 6. Unpublished work on the finite case was also done by
Boris Weisfeiler.
Subsequently,
Michael Collins Michael Collins or Mike Collins most commonly refers to:
* Michael Collins (Irish leader) (1890–1922), Irish revolutionary leader, soldier, and politician
* Michael Collins (astronaut) (1930–2021), American astronaut, member of Apollo 11 and Ge ...
, using the
classification of finite simple groups
In mathematics, the classification of the finite simple groups is a result of group theory stating that every finite simple group is either cyclic, or alternating, or it belongs to a broad infinite class called the groups of Lie type, or else i ...
, 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
References
{{DEFAULTSORT:Jordan-Schur Theorem
Theorems in group theory