In mathematical finite group theory, a quadratic pair for the odd
prime
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only way ...
''p'', introduced by , is a finite
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 ...
''G'' together with a quadratic module, a
faithful representation In mathematics, especially in an area of abstract algebra known as representation theory, a faithful representation ρ of a group on a vector space is a linear representation in which different elements of are represented by distinct linear ...
''M'' on a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
over the
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subt ...
with ''p'' elements such that ''G'' is generated by elements with
minimal polynomial (''x'' − 1)
2. Thompson classified the quadratic pairs for ''p'' ≥ 5. classified the quadratic pairs for ''p'' = 3. With a few exceptions, especially for ''p'' = 3, groups with a quadratic pair for the prime ''p'' tend to be more or less
groups of Lie type
In mathematics, specifically in group theory, the phrase ''group of Lie type'' usually refers to finite groups that are closely related to the group of rational points of a reductive linear algebraic group with values in a finite field. The phr ...
in characteristic ''p''.
See also
*
p-stable group
In finite group theory, a ''p''-stable group for an odd prime ''p'' is a finite group satisfying a technical condition introduced by in order to extend Thompson's uniqueness results in the odd order theorem to groups with dihedral Sylow 2-sub ...
References
*
*{{Citation , last1=Thompson , first1=John G. , author1-link=John G. Thompson , title=Actes du Congrès International des Mathématiciens (Nice, 1970) , publisher=Gauthier-Villars , mr=0430043 , year=1971 , volume=1 , chapter=Quadratic pairs , pages=375–376
Finite groups