General linear group
The general linear group or the degree over a field is the set of all regular matrices with coefficients from. Group join is the matrix multiplication. The name comes from the abbreviation of the English name "general linear group".
If the body is a finite field with a prime power, then we write also held. If the context is clear that the field of real or complex numbers is used as a basis, we also write or.
The general linear group and its subgroups are used in the representation of groups as well as in the study of symmetries.
General linear group on a vector space
If a vector space over a field is one or for the group of all automorphisms of, ie all bijective linear maps writes, with the sequential execution of such pictures as a group link.
If has finite dimension, and are isomorphic. For a given basis of the vector space can be represented by any automorphism by an invertible matrix. This is an isomorphism of on is made.
For the group is not abelian. For example, applies
The center consists of the multiples of the identity matrix ( with scalars from ).
Subgroups of GL ( n, k)
Each sub-group is called by a linear array. Some subgroups have special meaning.
- The subgroup of all diagonal matrices whose diagonal elements are all equal to 0, describes rescaling of space.
- Diagonal matrices that match all of the diagonal elements and are not 0, describe in geometry central dilations. The subset of these matrices is the center of. Only in the trivial case it is identical with.
- The special linear group consisting of all matrices with determinant 1 is a normal subgroup of; and the factor group is isomorphic to, the group of unit (without the 0).
- The orthogonal group contains all orthogonal matrices.
- The unitary group consists of all unitary matrices, ie those matrices whose adjoint is equal to its inverse. More generally, can the unitary group defined as a sub-group of linear transformations in a pre-Hilbert space, as well as the orthogonal group can be regarded as a subgroup of linear operators in a Euclidean vector space.
About the real and complex numbers
The general linear group over the field or is a Lie group over the field and has the dimension.
The Lie algebra to the Lie algebra General linear and consists of all the matrices with the commutator as the Lie bracket.
While is connected, has two connected components: the matrix of the positive and the negative determinant. The connected component with positive determinant contains the identity element and forms a subgroup. This subset is a connected Lie group with real dimension and has the same Lie algebra as.
Over finite fields
If a finite field having elements, then is a finite group of order
This value can be determined for example by counting the possibilities for the matrix columns: For the first column, there are possible occupancy (all except the zero column), for the second column there are ways (all except the multiples of the first column), etc.
Projective linear group
The projective linear group over a vector space over a field is the factor group, the normal ( even central ) subgroup of scalar multiples of the identity is made . The names, etc. are similar to those of the general linear group. If a finite field is, and are equally powerful, but in general not isomorphic.
The name is derived from the projective geometry, where the analogue of the general linear group is the projective linear group, the - dimensional projective space over this part of the group, it is the group of all projectivities of the room.
A special case is the group of Möbius transformations.
- Lie group
- Group Theory
- Linear Algebra