In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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-orie ...
''ƒ''(''n'') such that given a finite
subgroup
In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G.
Formally, given a group (mathematics), group under a binary operation ...
''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 iden ...
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 ...
''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 (: matrices or matrixes) or MATRIX may refer to:
Science and mathematics
* Matrix (mathematics), a rectangular array of numbers, symbols or expressions
* Matrix (logic), part of a formula in prenex normal form
* Matrix (biology), the ...
, there is a subgroup ''H'' of ''G'' with the following properties:
* ''H'' is
abelian
Abelian may refer to:
Mathematics Group theory
* Abelian group, a group in which the binary operation is commutative
** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms
* Metabelian group, a group ...
.
* ''H'' is a
normal subgroup
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group ...
of ''G''.
* The index of ''H'' in ''G'' satisfies (''G'' : ''H'') ≤ ''ƒ''(''n'').
Schur Schur is a German or Jewish surname. Notable people with the surname include:
* Alexander Schur (born 1971), German footballer
* Dina Feitelson-Schur (1926–1992), Israeli educator
* Friedrich Schur (1856-1932), German mathematician
* Fritz Schur ...
proved a more general result that applies when ''G'' is not assumed to be finite, but just
periodic
Periodicity or periodic may refer to:
Mathematics
* Bott periodicity theorem, addresses Bott periodicity: a modulo-8 recurrence relation in the homotopy groups of classical groups
* Periodic function, a function whose output contains values tha ...
. 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
In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number . It is denoted by (unrelated to the number ).
A symmetric variant seen sometimes is , which is equal ...
.
This was subsequently improved by
Hans Frederick Blichfeldt
Hans Frederick Blichfeldt (1873–1945) was a Danish-American mathematician at Stanford University, known for his contributions to group theory, the representation theory of finite groups, the geometry of numbers, sphere packing, and quadratic f ...
who replaced the 12 with a 6. Unpublished work on the finite case was also done by
Boris Weisfeiler
Boris Weisfeiler (born 19 April 1941 – disappeared 4–5 January 1985) was a Soviet-born mathematician and professor at Penn State University who lived in the United States before disappearing in Chile in 1985. Declassified US documents sugg ...
.
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 finite simple groups (popularly called the enormous theorem) is a result of group theory stating that every List of finite simple groups, finite simple group is either cyclic group, cyclic, or alternating gro ...
, 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
The Burnside problem asks whether a finitely generated group in which every element has finite order must necessarily be a finite group. It was posed by William Burnside in 1902, making it one of the oldest questions in group theory, and was inf ...
References
{{DEFAULTSORT:Jordan-Schur Theorem
Theorems in group theory