Groupprops, The Group Properties Wiki (pre-alpha)

From Groupprops

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

Contents 1 Definition 1.1 In terms of dimension 1.2 In terms of vector spaces 2 Particular cases 2.1 Finite fields

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 ^* )

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.

Particular cases

Finite fields

For q = 2, PSL(n,q) = SL(n,q) = PGL(n,q) = GL(n,q). For q a power of two, PGL(n,q) = PSL(n,q) = SL(n,q) but this is not the same as GL(n,q).

Size of field	Order of matrices	Common name for the projective special linear group
q	1	Trivial group
2	2	Symmetric group:S3
3	2	Symmetric group:S4
4	2	Alternating group:A5
5	2	Symmetric group:S5
9	2	Projective general linear group:PGL(2,9)
2	3	Projective special linear group:PSL(3,2)