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".
![\ mathrm {GL} (n, K)](https://wikimedia.org/api/rest_v1/media/math/render/svg/19dd1d99f5551a5d11bde5049faf4a7fa4cbad7e)
![\ mathrm {GL} _ {n} (K)](https://wikimedia.org/api/rest_v1/media/math/render/svg/b927823796448834198a1d33959863292d91e6b4)
![n \ times n](https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78)
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
![\ mathrm {GL}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f64ee3cbc41c3ea59b0915341e53a18149ee86e)
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 .
![q](https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d)
![\ mathrm {GL} (n, q)](https://wikimedia.org/api/rest_v1/media/math/render/svg/d23c35759b92db1d227c426d673398bcf92f9a2b)
![\ mathrm {GL} (n, K)](https://wikimedia.org/api/rest_v1/media/math/render/svg/19dd1d99f5551a5d11bde5049faf4a7fa4cbad7e)
![\ mathbb {R}](https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc)
![\ mathbb {C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7)
![\ mathrm {GL} (n)](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba99fb253ee3b9082e5d718da746260073e6c7b3)
![\ mathrm {GL} _ {n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b3b1b008d4ce8c41db8241a16ee90c7166217fbe)
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.
![V](https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845)
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
![\ mathrm {GL} (V)](https://wikimedia.org/api/rest_v1/media/math/render/svg/b9e9f3c8d0e7c42fb9ef98af46cf0c49f81eb1c9)
![\ mathrm {Aut} (V)](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c3f5e3555df7dcfdc0c3a37e3b9e2c97b6e5d54)
![V \ to V](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5ea4a1aff523f48f9e2d7ac4fb9c938d5cee7b8)
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 .
![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
![\ mathrm {GL} (V)](https://wikimedia.org/api/rest_v1/media/math/render/svg/b9e9f3c8d0e7c42fb9ef98af46cf0c49f81eb1c9)
![\ mathrm {GL} (n, K)](https://wikimedia.org/api/rest_v1/media/math/render/svg/19dd1d99f5551a5d11bde5049faf4a7fa4cbad7e)
![V](https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845)
![V](https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845)
![n \ times n](https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78)
![\ mathrm {GL} (V)](https://wikimedia.org/api/rest_v1/media/math/render/svg/b9e9f3c8d0e7c42fb9ef98af46cf0c49f81eb1c9)
![\ mathrm {GL} (n, K)](https://wikimedia.org/api/rest_v1/media/math/render/svg/19dd1d99f5551a5d11bde5049faf4a7fa4cbad7e)
For the group is not Abelian . For example,
![n \ geq 2](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6bf67f9d06ca3af619657f8d20ee1322da77174)
![n = 2](https://wikimedia.org/api/rest_v1/media/math/render/svg/a02c8bd752d2cc859747ca1f3a508281bdbc3b34)
![{\ begin {pmatrix} 1 & 0 \\ 1 & 1 \ end {pmatrix}} {\ begin {pmatrix} 1 & 1 \\ 0 & 1 \ end {pmatrix}} = {\ begin {pmatrix} 1 & 1 \\ 1 & 2 \ end {pmatrix}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/11693ab12978685e4996a1dd0df89b069388b047)
but
-
.
The center of consists of the multiples of the identity matrix (with scalars off ).
![\ mathrm {GL} (n, K)](https://wikimedia.org/api/rest_v1/media/math/render/svg/19dd1d99f5551a5d11bde5049faf4a7fa4cbad7e)
![K \ setminus \ {0 \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f12835c964f389ba3df759462067bf0087becac7)
Subgroups of GL (n, K)
Each subgroup of is called a matrix group or linear group . Some subgroups have special meaning.
![\ mathrm {GL} (n, K)](https://wikimedia.org/api/rest_v1/media/math/render/svg/19dd1d99f5551a5d11bde5049faf4a7fa4cbad7e)
- 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.
![\ mathrm {GL} (n, K)](https://wikimedia.org/api/rest_v1/media/math/render/svg/19dd1d99f5551a5d11bde5049faf4a7fa4cbad7e)
![n = 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/d9ec7e1edc2e6d98f5aec2a39ae5f1c99d1e1425)
![\ mathrm {GL} (n, K)](https://wikimedia.org/api/rest_v1/media/math/render/svg/19dd1d99f5551a5d11bde5049faf4a7fa4cbad7e)
- 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).
![\ mathrm {SL} (n, K)](https://wikimedia.org/api/rest_v1/media/math/render/svg/4dd77477c38e36ab0e5e1ea4453ce8ccc040be73)
![\ mathrm {SL} (n, K)](https://wikimedia.org/api/rest_v1/media/math/render/svg/4dd77477c38e36ab0e5e1ea4453ce8ccc040be73)
![\ mathrm {GL} (n, K) / \ mathrm {SL} (n, K)](https://wikimedia.org/api/rest_v1/media/math/render/svg/570c32eac288c68dc8081d640cb104870aec7f3a)
![K ^ {\ times}](https://wikimedia.org/api/rest_v1/media/math/render/svg/98403ab9e2f27fde4576ca3e622add252f4fd9a4)
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
- The orthogonal group contains all orthogonal matrices .
![\ mathrm {O} (n, K)](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc16d0ff893530ff3e3cb30d6a35686f6b8eb072)
- For these matrices describe automorphisms of which contain the Euclidean norm and the scalar product , i.e. orthogonal mappings .
![K = \ mathbb {R}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6419d3aa99701ca996737b17a5e1174d53e6c9e)
![\ mathbb {R} ^ {n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d)
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 .
![\ mathrm {GL} (n)](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba99fb253ee3b9082e5d718da746260073e6c7b3)
![\ mathbb {R}](https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc)
![\ mathbb {C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7)
![n ^ {2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac9810bbdafe4a6a8061338db0f74e25b7952620)
- 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 .![\ mathrm {Mat} _ {n} (K)](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d7d0b4fef2d6e1346780f86ea0fa42f41b696c4)
![n \ times n](https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78)
![n ^ {2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac9810bbdafe4a6a8061338db0f74e25b7952620)
![\ mathrm {Mat} _ {n} (K) \ \ rightarrow \ K](https://wikimedia.org/api/rest_v1/media/math/render/svg/91824ac89827aaaed3c007001dce60347d2655f6)
![\ mathrm {GL} (n)](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba99fb253ee3b9082e5d718da746260073e6c7b3)
![K ^ {\ times}](https://wikimedia.org/api/rest_v1/media/math/render/svg/98403ab9e2f27fde4576ca3e622add252f4fd9a4)
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
![\ mathrm {Mat} _ {n} (K)](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d7d0b4fef2d6e1346780f86ea0fa42f41b696c4)
![n ^ {2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac9810bbdafe4a6a8061338db0f74e25b7952620)
The Lie algebra zu is the general linear Lie algebra . This consists of all matrices with the commutator as a Lie bracket .
![{\ mathfrak {gl}} (n)](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7ec46e55b384a843fea7a4532b53f1504cf04de)
![n \ times n](https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78)
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 .
![\ mathrm {GL} (n, \ mathbb {R})](https://wikimedia.org/api/rest_v1/media/math/render/svg/c32cfbbae2e2ae75b6647a2bda87942637fe4bc1)
![\ mathrm {GL} ^ {+} (n, \ mathbb {R})](https://wikimedia.org/api/rest_v1/media/math/render/svg/c47b24968806c6c54f9d3113d28bf07139744cb2)
![n ^ {2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac9810bbdafe4a6a8061338db0f74e25b7952620)
![\ mathrm {GL} (n, \ mathbb {R})](https://wikimedia.org/api/rest_v1/media/math/render/svg/c32cfbbae2e2ae75b6647a2bda87942637fe4bc1)
Over finite bodies
If is a finite field with elements, where is a prime, then is a finite group of order
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
![p](https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36)
![p](https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36)
![{\ displaystyle \ mathrm {GL} (n, p)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7cb6261478c1fa205c511d07102678f212924aa3)
![{\ displaystyle \ prod _ {i = 0} ^ {n-1} \ left (p ^ {n} -p ^ {i} \ right) = \ left (p ^ {n} -1 \ right) \ cdot \ left (p ^ {n} -p \ right) \ cdot \ left (p ^ {n} -p ^ {2} \ right) \ cdots \ left (p ^ {n} -p ^ {n-1} \ right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a7145575a9be442aa94aaee2a7984a4fc4f2a38)
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.
![{\ displaystyle p ^ {n} -1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7906fcb5c1c777d94606871f50bbd283ca474aca)
![{\ displaystyle p ^ {n} -p}](https://wikimedia.org/api/rest_v1/media/math/render/svg/48a752043e0c6d2fa1dcc9a29e46c5d72cb8f6e8)
If is a finite field with elements, where is a prime, then is a finite group of order
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
![{\ displaystyle q = p ^ {k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9535c3e1dfaf62cc46386c4a0d8aabf56b40699a)
![p](https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36)
![{\ displaystyle \ mathrm {GL} (n, q)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d23c35759b92db1d227c426d673398bcf92f9a2b)
![{\ displaystyle \ prod _ {i = 0} ^ {n-1} \ left (q ^ {n} -q ^ {i} \ right) = \ left (q ^ {n} -1 \ right) \ cdot \ left (q ^ {n} -q \ right) \ cdot \ left (q ^ {n} -q ^ {2} \ right) \ cdots \ left (q ^ {n} -q ^ {n-1} \ right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca0ddc85e7493f224afbfd2940dfc711a333e72d)
Note: Above the ring with elements, where is prime, the group is a finite group of order
![{\ displaystyle \ mathbb {Z} _ {p ^ {k}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f03c2e5f6460975387ab4f88a0d54ca72b2f966)
![p ^ {k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e017c102135ab13bdf501dc1c1b5fd1840a97822)
![p](https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36)
![{\ displaystyle \ mathrm {GL} (n, \ mathbb {Z} _ {p ^ {k}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb3c7c5578df4bf6694ebb5d8620426940cc405d)
![{\ displaystyle p ^ {(k-1) n ^ {2}} \ prod _ {i = 0} ^ {n-1} \ left (p ^ {n} -p ^ {i} \ right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c44cfc1067014a53003881bf82d4df565e47eaf8)
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.
![\ mathrm {PGL} (V)](https://wikimedia.org/api/rest_v1/media/math/render/svg/c43d040dbb415ed5f5e48f37e0e63049045aeee0)
![V](https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845)
![\ mathrm {GL} (V) / K ^ {\ times}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bdbb2f0faac4d1a6e7b7d6d7ecc0699f2c4bfd09)
![K ^ {\ times}](https://wikimedia.org/api/rest_v1/media/math/render/svg/98403ab9e2f27fde4576ca3e622add252f4fd9a4)
![\ mathrm {id}: V \ rightarrow V](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a1fb2f5c03f74c6b6f53ed6b5aa5ac6a8950515)
![k](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40)
![K \ setminus \ {0 \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f12835c964f389ba3df759462067bf0087becac7)
![\ mathrm {PGL} (n, K)](https://wikimedia.org/api/rest_v1/media/math/render/svg/424ffc98f4480abc233fd7db8ac6c562475b8f69)
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
![\ mathrm {PGL} (n, K)](https://wikimedia.org/api/rest_v1/media/math/render/svg/424ffc98f4480abc233fd7db8ac6c562475b8f69)
![\ mathrm {SL} (n, K)](https://wikimedia.org/api/rest_v1/media/math/render/svg/4dd77477c38e36ab0e5e1ea4453ce8ccc040be73)
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.
![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
![\ mathrm {PGL} (n + 1, K)](https://wikimedia.org/api/rest_v1/media/math/render/svg/46bdb258308514d84cc48cbffac138da93880cfa)
A special case is the group of Möbius transformations , the .
![\ mathrm {PGL} (2, \ mathbb {C})](https://wikimedia.org/api/rest_v1/media/math/render/svg/482bf5010a37239ce2aa6dbbe445905c4138f9b8)
Web links
Individual evidence
-
↑ Jeffrey Overbey, William Traves and Jerzy Wojdylo: On The Keyspace Of The Hill Cipher . (PDF).