General linear group over a field
This article defines a natural number-parametrized system of algebraic matrix groups. In other words, for every field and every natural number, we get a matrix group defined by a system of algebraic equations. The definition may also generalize to arbitrary commutative unital rings, though the default usage of the term is over fields.
View other linear algebraic groups|View other affine algebraic groups
- 1 Definition
- 2 More general versions
- 3 As a group with additional structure
- 4 As a map
- 5 Relation with other linear algebraic groups
- 6 Subgroups
- 7 Quotients
- 8 Supergroups
- 9 Particular cases
- 10 GAP implementation
In terms of dimension (finite-dimensional case)
Let be a natural number and a field. The general linear group of degree over , denoted , is defined in the following equivalent ways:
- is the group of all invertible -linear maps from the vector space to itself, under composition. In other words, it is the group of automorphisms of as a -vector space.
- is the group of all invertible matrices with entries over
In terms of vector spaces
Let be a -vector space (which may be finite or infinite-dimensional). The general linear group over , denoted , is the group of all vector space automorphisms from to itself.
Note that when , this reduces to the definition . Further, since for , and since any two vector spaces of the same dimension are isomorphic, the s cover all general linear groups corresponding to finite-dimensional vector spaces.
More general versions
Some more general versions occur where we replace field by more general constructs:
|Replacement of field||Corresponding analogue of general linear group|
|division ring||general linear group over a division ring|
|commutative unital ring||general linear group over a commutative unital ring|
|unital ring||general linear group over a unital ring|
As a group with additional structure
As an algebraic group
Further information: general linear group over a field as an algebraic group
For any field , the general linear group has the natural structure of an algebraic group. It can be viewed as a Zariski-open subvariety in given by the condition that the determinant function does not vanish.
As a Lie group
As a map
As a functor from fields to groups
If we fix , we can think of as a functor from the category of fields to the category of groups.
As an IAPS
Further information: GL IAPS
For a fixed field , the general linear groups form an IAPS of groups parametrized by . In other words, we naturally have concatenation maps:
This map takes a matrix of order and a matrix of order and putputs the block diagonal matrix with blocks and .
As a functor from fields to IAPSes
If we fix neither nor , we can view as a functor from fields to the category of IAPSes of groups.
Relation with other linear algebraic groups
The general linear group has a number of important subgroups. Some of them are listed below:
- Orthogonal group is the subgroup comprising those matrices such that is the identity matrix. Further information: Orthogonal group in general linear group
- Special linear group is the subgroup comprising those matrices with determinant 1. Further information: Special linear group in general linear group
- Symplectic group is the subgroup comprising matrices such that where is a certain fixed matrix. Further information: Symplectic group in general linear group
- Special orthogonal group is the intersection of the orthogonal group and the special linear group
- Orthogonal similitude group is the group of matrices such that is a scalar matrix
- Symmetric group which is the group of permutation matrices, embedded in the general linear group
All of these, except the orthogonal similtude group, form sub-IAPSes.
Inner automorphism group
Some important groups in which the general linear group is contained:
- General affine group which is the semidirect product of the vector space (as an Abelian group) and the general linear group acting on it
- Skew-linear group which is the semidirect product of the general linear group with the transpose-inverse map
Here are some general facts:
- Multiplicative group of a finite field is cyclic
- Special linear group is quasisimple for , except the case that and has two or three elements. Thus, all the corresponding general linear groups have a simple non-abelian subquotient. In the case that the field has characteristic two, the general linear group coincides with the special linear group, and it is centerless, so it turns out to be a simple non-abelian group itself.
|Size of field||Order of matrices||Common name for the general linear group||Order of group||Comment|
|3||1||Cyclic group:Z2||group of prime order|
|4||1||Cyclic group:Z3||group of prime order|
|5||1||Cyclic group:Z4||cyclic group|
|2||2||Symmetric group:S3||supersolvable but not nilpotent|
|3||2||General linear group:GL(2,3)||solvable but not supersolvable|
|4||2||Alternating group:A5||simple non-abelian group|
|5||2||General linear group:GL(2,5)||not solvable, has a simple non-abelian subquotient.|
|2||3||General linear group:GL(3,2)||simple non-abelian group|
The general linear group can be implemented using the GAP function GeneralLinearGroup that can be invoked as either GeneralLinearGroup or GL. It takes two arguments, the first being the degree (i.e., the order of matrices) and the second either a ring or a prime power (for which we consider the corresponding field). For instance, for the general linear group of degree two over the field of three elements:
G := GL(2,3);