General linear group
The general linear group or the degree over a body is the group of all regular - matrices made with coefficients . Group linking is matrix multiplication . The name comes from generally linear or the English term " g eneral l inear group".
If the body is a finite body with a prime power , one also writes instead . If it is clear from the context that the body of the real or the complex numbers is taken as a basis, one writes or .
The general linear group and its subgroups are used in the representation of groups as well as in the study of symmetries .
Subgroups of the general linear group are called matrix groups .
General linear group over a vector space
If a vector space is over a body , one writes or for the group of all automorphisms of , i.e. all bijective linear mappings , with the execution of such mappings one after the other as a group connection.
If the finite dimension has are and isomorphic. For a given basis of vector space , each automorphism of can be represented by an invertible matrix. This creates an isomorphism from to .
For the group is not Abelian . For example,
but
- .
The center of consists of the multiples of the identity matrix (with scalars off ).
Subgroups of GL (n, K)
Each subgroup of is called a matrix group or linear group . Some subgroups have special meaning.
- The subgroup of all diagonal matrices , whose diagonal elements are all not equal to 0, describes rescaling of the space.
- Diagonal matrices, in which all diagonal elements match and are not 0, describe centric extensions in geometry . The subset of these matrices is the center of . Only in the trivial case it is identical.
- The special linear group consists of all matrices with the determinant 1. is a normal divisor of ; and the factor group is isomorphic to , the unit group of (without the 0).
- The orthogonal group contains all orthogonal matrices .
- For these matrices describe automorphisms of which contain the Euclidean norm and the scalar product , i.e. orthogonal mappings .
- The unitary group consists of all unitary matrices , that is, those matrices whose adjoint is equal to its inverse . More generally, the unitary group can be defined as a subgroup of the linear mappings in a Prehilbert space , just as the orthogonal group can be understood as a subgroup of the linear mappings in a Euclidean vector space .
- The affine group AGL _{n} (K) is a subgroup of .
About the real and complex numbers
The general linear group over the body or is an algebraic group and thus in particular a Lie group over the body and has the dimension .
- Proof:
- is a subset of the manifold of all matrices, which is a vector space of dimension . The determinant is a polynomial and therefore especially a continuous mapping . As the archetype of the open subset of is an open, not empty subset of and therefore also has the dimension .
The Lie algebra zu is the general linear Lie algebra . This consists of all matrices with the commutator as a Lie bracket .
While connected is has two connected components: the matrices with positive and negative determinant. The connected component with a positive determinant contains the unit element and forms a subgroup . This subgroup is a connected Lie group with real dimensions and has the same Lie algebra as .
Over finite bodies
If is a finite field with elements, where is a prime, 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 assignment options (all except the zero column), for the second column there are options (all except the multiples of the first column) etc.
If is a finite field with elements, where is a prime, then is a finite group of order
Note: Above the ring with elements, where is prime, the group is a finite group of order
There are some peculiarities for the general linear group over the body with 2 elements. First of all they coincide with the projective and special projective groups , that is
- .
In particular, these groups are for simple and in small dimensions exist the following isomorphisms:
- , that is the symmetrical group S _{3} with 6 elements
- , that is the simple group with 168 elements
- , that is the alternating group A _{8} with 20160 elements.
Projective linear group
The projective linear group over a vector space over a field is the factor group , with the normal (even central ) subgroup of the scalar multiples of identity being with off . The names etc. correspond to those of the general linear group. If a finite body are and equally powerful but generally not isomorphic.
The name comes from projective geometry , where the analogue to the general linear group is the projective linear group , while the group belongs to the -dimensional projective space , it is the group of all projectivities of space.
A special case is the group of Möbius transformations , the .
Web links
- General linear group in the Encyclopaedia of Mathematics
- Eric W. Weisstein : General Linear Group . In: MathWorld (English).
Individual evidence
- ↑ Jeffrey Overbey, William Traves and Jerzy Wojdylo: On The Keyspace Of The Hill Cipher . (PDF).