|
gptkbp:instanceOf
|
gptkb:group_of_people
gptkb:Lie_group
|
|
gptkbp:actsOn
|
vector space of dimension n over F
|
|
gptkbp:application
|
gptkb:algebraic_geometry
cryptography
differential geometry
group theory
number theory
physics
representation theory
linear algebra
|
|
gptkbp:automorphismGroup
|
vector space of dimension n over F
|
|
gptkbp:centralTo
|
scalar matrices
|
|
gptkbp:containsElement
|
invertible n x n matrices over F
|
|
gptkbp:determinant
|
nonzero
GL(n, F) → F^*
|
|
gptkbp:dimensions
|
n^2
|
|
gptkbp:field
|
F
|
|
gptkbp:fullName
|
gptkb:general_linear_group_of_degree_n_over_field_F
|
|
gptkbp:generation
|
elementary matrices
|
|
gptkbp:hasConnection
|
yes (if F is algebraically closed)
|
|
gptkbp:hasDeterminantMap
|
yes
|
|
gptkbp:hasSubgroup
|
gptkb:SL(n,_F)
GL(m, F) for m > n
|
|
https://www.w3.org/2000/01/rdf-schema#label
|
GL(n, F)
|
|
gptkbp:identityElement
|
identity matrix
|
|
gptkbp:isAlgebraicGroup
|
yes
|
|
gptkbp:isClassicalGroup
|
yes
|
|
gptkbp:isFinite
|
yes (if F is finite)
|
|
gptkbp:isMatrixGroup
|
yes (if F = R or C)
|
|
gptkbp:isNonAbelian
|
yes (for n > 1)
|
|
gptkbp:isQuotientOf
|
gptkb:PGL(n,_F)
|
|
gptkbp:isReductive
|
yes
|
|
gptkbp:isSimple
|
no (for n > 1, except in some cases)
|
|
gptkbp:isTopologicalGroup
|
yes (if F is topological field)
|
|
gptkbp:kernelOfDeterminantMap
|
gptkb:SL(n,_F)
|
|
gptkbp:notation
|
gptkb:GL(n,_F)
|
|
gptkbp:operator
|
matrix multiplication
|
|
gptkbp:order
|
finite (if F is finite)
infinite (if F is infinite)
product_{k=0}^{n-1} (|F|^n - |F|^k) (if F is finite)
|
|
gptkbp:relatedTo
|
orthogonal group
projective linear group
|
|
gptkbp:representationTheory
|
well-studied
|
|
gptkbp:bfsParent
|
gptkb:general_linear_group
gptkb:orthogonal_group
|
|
gptkbp:bfsLayer
|
5
|