Projective general linear group
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
In terms of dimension
- It is the group of automorphisms of projective space of dimension , that arise from linear automorphisms of the vector space of dimension .
- It is the quotient of by its center, viz the group of scalar multiplies of the identity (isomorphic to the group )
For a prime power, we denote by the group where is the field (unique up to isomorphism) of size .
In terms of vector spaces
Let be a vector space over a field . The projective general linear group of , denoted , is defined as the inner automorphism group of , viz the quotient of by its center, which is the group of scalar multiples of the identity transformation.
Over a finite field
Below is information for the projective general linear group of degree over a finite field of size .
|order||has the same order||The order of is . The center has order , so the quotient has order equal to the quotient of the order of by .|
|number of conjugacy classes||(no good generic expression, but overall it's a polynomial in of degree )||See element structure of projective general linear group over a finite field|
Particular cases by degree
|Degree||Information on projective general linear group over a field|
|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|
If , then for all .
More generally, if is relatively prime to , then the groups are all isomorphic to each other. However, they are not isomorphic to .
|Size of field||Characteristic of field (so is a power of||Degree of projective general linear group||Common name for the projective general linear group||Order of|
|9||3||2||Projective general linear group:PGL(2,9)||720|
|2||2||3||Projective special linear group:PSL(3,2)||168|