Burnside theorem
Encyclopedia
In mathematics
, Burnside's theorem in group theory
states that if G is a finite group
of order
where p and q are prime number
s, and a and b are non-negative integer
s, then G is solvable
. Hence each
non-Abelian finite simple group has order divisible by three distinct primes. Furthermore, as a consequence of the Feit-Thompson theorem, one of those can be chosen to be 2.
in the early years of the 20th century.
Burnside's theorem has long been one of the best-known applications of representation theory
to the theory of finite groups, though a proof avoiding the use of group characters was published by D. Goldschmidt around 1970.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Burnside's theorem in group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
states that if G is a finite group
Finite group
In mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of...
of order
Order (group theory)
In group theory, a branch of mathematics, the term order is used in two closely related senses:* The order of a group is its cardinality, i.e., the number of its elements....
where p and q are prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
s, and a and b are non-negative integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s, then G is solvable
Solvable group
In mathematics, more specifically in the field of group theory, a solvable group is a group that can be constructed from abelian groups using extensions...
. Hence each
non-Abelian finite simple group has order divisible by three distinct primes. Furthermore, as a consequence of the Feit-Thompson theorem, one of those can be chosen to be 2.
History
The theorem was proved by William BurnsideWilliam Burnside
William Burnside was an English mathematician. He is known mostly as an early contributor to the theory of finite groups....
in the early years of the 20th century.
Burnside's theorem has long been one of the best-known applications of representation theory
Representation theory of finite groups
In mathematics, representation theory is a technique for analyzing abstract groups in terms of groups of linear transformations. See the article on group representations for an introduction...
to the theory of finite groups, though a proof avoiding the use of group characters was published by D. Goldschmidt around 1970.
Outline of Burnside's proof
- By induction, it suffices to prove that a finite simple group G whose order has the form
for primes p and q is cyclic. Suppose then that the order of G has this form, but G is not cyclic. Suppose for definiteness that b >0. - Using the modified class equation, G has a non-identity conjugacy class of size prime to q. Hence G either has a non-trivial center, or has a conjugacy classConjugacy classIn mathematics, especially group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes of non-abelian groups reveals many important features of their structure...
of size
for some positive integer r. The first possibility is excluded since G is assumed simple, but not cyclic. Hence there is a non-central element x of G such that the conjugacy class of x has size 
. - Application of column orthogonality relationsCharacter theoryIn mathematics, more specifically in group theory, the character of a group representation is a function on the group which associates to each group element the trace of the corresponding matrix....
and other properties of group characters and algebraic integers lead to the existence of a non-trivial irreducible characterCharacter theoryIn mathematics, more specifically in group theory, the character of a group representation is a function on the group which associates to each group element the trace of the corresponding matrix....
of G such that 
. - The simplicity of G then implies that any non-trivial complex irreducible representation is faithful, and it follows that x is in the center of G, a contradiction.