Projective general linear group

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This term associates to every field, a corresponding group property. In other words, given a field, every group either has the property with respect to that field or does not have the property with respect to that field

This group property is natural number-parametrized, in other words, for every natural number, we get a corresponding group property

Definition

In terms of dimension

Let n be a natural number and k be a field. The projective general linear group of order n over k, denoted PGL(n,k) is defined in the following equivalent ways:

  • It is the group of automorphisms of projective space of dimension n-1, that arise from linear automorphisms of the vector space of dimension n.
  • It is the quotient of GL(n,k) by its center, viz the group of scalar multiplies of the identity (isomorphic to the group k^*)

For q a prime power, we denote by PGL(n,q) the group PGL(n,\mathbb{F}_q) where \mathbb{F}_q is the field (unique up to isomorphism) of size q.

In terms of vector spaces

Let V be a vector space over a field k. The projective general linear group of V, denoted PGL(V), is defined as the inner automorphism group of GL(V), viz the quotient of GL(V) by its center, which is the group of scalar multiples of the identity transformation.

Arithmetic functions

Over a finite field

Below is information for the projective general linear group of degree n over a finite field of size q.

Function Value Similar groups Explanation
order q^{\binom{n}{2}}\prod_{i=2}^n (q^i - 1) SL(n,q) has the same order The order of GL(n,q) is q^{\binom{n}{2}}\prod_{i=1}^n (q^i - 1). The center has order q - 1, so the quotient has order equal to the quotient of the order of GL(n,q) by q - 1.
number of conjugacy classes (no good generic expression, but overall it's a polynomial in q of degree n - 1) See element structure of projective general linear group over a finite field

Particular cases

Particular cases by degree

Degree n Information on projective general linear group PGL(n,k) over a field k
1 trivial group always
2 projective general linear group of degree two
3 projective general linear group of degree three
4 projective general linear group of degree four

Finite fields

If q = 2, then PGL(n,q) = GL(n,q) = SL(n,q) = PSL(n,q) for all n.

More generally, if q - 1 is relatively prime to n, then the groups PGL(n,q), SL(n,q), PSL(n,q) are all isomorphic to each other. However, they are not isomorphic to GL(n,q).

Size of field q Characteristic p of field (so q is a power of p Degree of projective general linear group n Common name for the projective general linear group PGL(n,q) = PGL(n,\mathbb{F}_q) Order of PGL(n,q)
q p 1 Trivial group 1
2 2 2 Symmetric group:S3 6
3 3 2 Symmetric group:S4 24
4 2 2 Alternating group:A5 60
5 5 2 Symmetric group:S5 120
9 3 2 Projective general linear group:PGL(2,9) 720
2 2 3 Projective special linear group:PSL(3,2) 168