Quadratic pair
Encyclopedia
In mathematical finite group theory, a quadratic pair for the odd prime
p, introduced by , is a finite group
G together with a quadratic module, a faithful representation
M on a vector space
over the finite field
with p elements such that G is generated by elements with minimum 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 characteristic p.
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...
p, introduced by , is 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 together with a quadratic module, a faithful representation
Faithful representation
In mathematics, especially in the area of abstract algebra known as representation theory, a faithful representation ρ of a group G on a vector space V is a linear representation in which different elements g of G are represented by distinct linear mappings ρ.In more abstract language, this means...
M on a vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
over the finite field
Finite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
with p elements such that G is generated by elements with minimum 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 characteristic p.