The Construction of a Matrix Group
Construction of the General Linear Group
GeneralLinearGroup(n, R) : RngIntElt, Rng -> GrpMat
Example GrpMatGen_Create (H59E1)
Construction of a Matrix Group Element
elt< G | L > : GrpMat, List(RngElt) -> GrpMatElt
G ! Q : GrpMat, [ RngElt ] -> GrpMatElt
ElementToSequence(g) : GrpMatElt -> [ RngElt ]
Identity(G) : GrpMat -> GrpMatElt
Example GrpMatGen_Matrices (H59E2)
Construction of a General Matrix Group
MatrixGroup< n, R | L > : RngIntElt, Rng, List -> GrpMat
Example GrpMatGen_Constructor (H59E3)
Example GrpMatGen_GLSylow (H59E4)
Changing Rings
ChangeRing(G, S) : GrpMat, Rng -> GrpMat, Map
ChangeRing(G, S, f) : GrpMat, Rng, Map -> GrpMat, Map
RestrictField(G, S) : GrpMat, FldFin -> GrpMat, Map
ExtendField(G, L) : GrpMat, FldFin -> GrpMat, Map
Coercion between Matrix Structures
R ! g : AlgMat, GrpMatElt -> RngMatElt
G ! r : GrpMat, AlgMatElt -> GrpMatElt
M ! g : ModMatRng, GrpMatElt -> ModMatRngElt
G ! m : GrpMat, ModMatRngElt -> GrpMatElt
Accessing Associated Structures
G . i : GrpMat, RngIntElt -> GrpMatElt
Degree(G) : GrpMat -> RngIntElt
Generators(G) : GrpMat -> { GrpMatElt }
NumberOfGenerators(G) : GrpMat -> RngIntElt
CoefficientRing(G) : GrpMat -> Rng
RSpace(G) : GrpMat -> ModTupRng
VectorSpace(G) : GrpMat -> ModTupFld
GModule(G) : GrpMat -> ModGrp
Generic(G) : GrpMat -> GrpMat
Parent(G) : GrpMatElt -> GrpMat
Homomorphisms
hom<G -> H | L> : GrpMat, Grp, List -> Map
Domain(f) : Map -> Grp
Codomain(f) : Map -> Grp
Image(f) : Map -> Grp
Kernel(f) : Map -> Grp
IsHomomorphism(G, H, Q) : GrpMat, GrpMat, SeqEnum[GrpMatElt] -> Bool, Map
Example GrpMatGen_Homomorphism (H59E5)
Construction of Extensions
DirectProduct(G, H) : GrpMat, GrpMat -> GrpMat
DirectProduct(Q) : [ GrpMat ] -> GrpMat
SemiLinearGroup(G, S) : GrpMat, FldFin -> GrpMat
TensorWreathProduct(G, H) : GrpMat, GrpPerm -> GrpMat
WreathProduct(G, H) : GrpMat, GrpPerm -> GrpMat
Example GrpMatGen_Constructions (H59E6)
Arithmetic with Matrices
g * h : GrpMatElt, GrpMatElt -> GrpMatElt
g ^ n : GrpMatElt, RngIntElt -> GrpMatElt
g / h : GrpMatElt, GrpMatElt -> GrpMatElt
g ^ h : GrpMatElt, GrpMatElt -> GrpMatElt
(g, h) : GrpMatElt, GrpMatElt -> GrpMatElt
(g1, ..., gr) : GrpMatElt, ..., GrpMatElt -> GrpMatElt
Example GrpMatGen_Arithmetic (H59E7)
Predicates for Matrices
g eq h : GrpMatElt, GrpMatElt -> BoolElt
g ne h : GrpMatElt, GrpMatElt -> BoolElt
IsIdentity(g) : GrpMatElt -> BoolElt
IsScalar(g) : GrpMatElt -> BoolElt
Matrix Invariants
Degree(g) : GrpMatElt -> RngIntElt
HasFiniteOrder(g) : GrpMatElt -> BoolElt, RngIntElt
Order(g) : GrpMatElt -> RngIntElt, BoolElt
FactoredOrder(g) : GrpMatElt -> [ <RngIntElt, RngIntElt> ], BoolElt
ProjectiveOrder(g) : GrpMatElt -> RngIntElt, RngElt
FactoredProjectiveOrder(A) : AlgMatElt -> [ <RngIntElt, RngIntElt> ], RngElt
CentralOrder(g : parameters) : GrpMatElt -> RngIntElt, BoolElt
Determinant(g) : GrpMatElt -> RngElt
Trace(g) : GrpMatElt -> RngElt
CharacteristicPolynomial(g: parameters) : GrpMatElt -> RngPolElt
MinimalPolynomial(g) : GrpMatElt -> RngPolElt
Example GrpMatGen_Invariants (H59E8)
Group Order
IsFinite(G) : GrpMat -> Bool, RngIntElt
Order(G) : GrpMat -> RngIntElt
FactoredOrder(G) : GrpMat -> [ <RngIntElt, RngIntElt> ]
Example GrpMatGen_Order (H59E9)
Membership and Equality
g in G : GrpMatElt, GrpMat -> BoolElt
g notin G : GrpMatElt, GrpMat -> BoolElt
S subset G : { GrpMatElt }, GrpMat -> BoolElt
H subset G : GrpMat, GrpMat -> BoolElt
S notsubset G : { GrpMatElt }, GrpMat -> BoolElt
H notsubset G : GrpMat, GrpMat -> BoolElt
H eq G : GrpMat, GrpMat -> BoolElt
H ne G : GrpMat, GrpMat -> BoolElt
Set Operations
NumberingMap(G) : GrpMat -> Map
RandomProcess(G) : GrpMat -> Process
Random(G: parameters) : GrpMat -> GrpMatElt
Random(P) : Process -> GrpMatElt
Example GrpMatGen_Random (H59E10)
Abstract Group Predicates
IsAbelian(G) : GrpMat -> BoolElt
IsCyclic(G) : GrpMat -> BoolElt
IsElementaryAbelian(G) : GrpMat -> BoolElt
IsNilpotent(G) : GrpMat -> BoolElt
IsSoluble(G) : GrpMat -> BoolElt
IsPerfect(G) : GrpMat -> BoolElt
IsSimple(G) : GrpMat -> BoolElt
Example GrpMatGen_Order (H59E11)
Conjugacy
Class(H, x) : GrpMat, GrpMatElt -> { GrpMatElt }
ClassMap(G) : GrpMat -> Map
ConjugacyClasses(G: parameters) : GrpMat -> [ < RngIntElt, RngIntElt, GrpMatElt > ]
ClassRepresentative(G, x) : GrpMat, GrpMatElt -> GrpMatElt
ClassCentraliser(G, i) : GrpMat, RngIntElt -> GrpMat
ClassInvariants(G, g) : GrpMat, GrpMatElt -> .
ClassRepresentativeFromInvariants(G, p, h, t) : GrpMat, SeqEnum, SeqEnum, FldFinElt -> GrpMatElt
IsConjugate(G, g, h) : GrpMat, GrpMatElt, GrpMatElt -> BoolElt, GrpMatElt | Unass
IsConjugate(G, H, K) : GrpMat, GrpMat, GrpMat -> BoolElt, GrpMatElt | Unass
IsGLConjugate(H, K) : GrpMat, GrpMat -> BoolElt, GrpMatElt | Unass
Exponent(G) : GrpMat -> RngIntElt
NumberOfClasses(G) : GrpMat -> RngIntElt
PowerMap(G) : GrpMat -> Map
AssertAttribute(G, "Classes", Q) : GrpMat, MonStgElt, SeqEnum ->
Example GrpMatGen_RationalMatrixGroupDatabase (H59E12)
Construction of Subgroups
sub<G | L> : GrpMat, List -> GrpMat
ncl<G | L> : GrpMat, List -> GrpMat
Example GrpMatGen_Subgroups (H59E13)
Elementary Properties of Subgroups
Index(G, H) : GrpMat, GrpMat -> RngIntElt
FactoredIndex(G, H) : GrpMat, GrpMat -> [ <RngIntElt, RngIntElt> ]
IsCentral(G, H) : GrpMat -> BoolElt
IsMaximal(G, H) : GrpMat, GrpMat -> BoolElt
IsNormal(G, H) : GrpMat, GrpMat -> BoolElt
IsSubnormal(G, H) : GrpMat, GrpMat -> BoolElt
Standard Subgroups
H ^ g : GrpMat, GrpMatElt -> GrpMat
H meet K : GrpMat, GrpMat -> GrpMat
CommutatorSubgroup(G, H, K) : GrpMat, GrpMat, GrpMat -> GrpMat
Centralizer(G, g) : GrpMat, GrpMatElt -> GrpMat
Centralizer(G, H) : GrpMat, GrpMat -> GrpMat
Core(G, H) : GrpMat, GrpMat -> GrpMat
H ^ G : GrpMat, GrpMat -> GrpMat
Normalizer(G, H) : GrpMat, GrpMat -> GrpMat
SylowSubgroup(G, p) : GrpMat, RngIntElt -> GrpMat
pCore(G, p) : GrpMat, RngIntElt -> GrpMat
Low Index Subgroups
LowIndexSubgroups(G, R : parameters) : GrpMat, RngIntElt -> [ GrpMat ]
Example GrpMatGen_LowIndexMatrixGroup (H59E14)
Conjugacy Classes of Subgroups
SubgroupClasses(G: parameters) : GrpMat -> [ rec< GrpMat, RngIntElt, RngIntElt, GrpFP> ]
MaximalSubgroups(G: parameters) : GrpMat -> [ rec< GrpMat, RngIntElt, RngIntElt, GrpFP> ]
SubgroupsLift(G, A, B, Q: parameters) : GrpMat, GrpMat, GrpMat, SeqEnum -> SeqEnum
Construction of Quotient Groups
quo<G | L> : GrpMat, List -> GrpPerm, Map
G / N : GrpMat, GrpMat -> GrpPerm
Example GrpMatGen_Quotient (H59E15)
Abelian, Nilpotent and Soluble Quotients
AbelianQuotient(G) : GrpMat -> GrpAb, Map
ElementaryAbelianQuotient(G, p) : GrpMat, RngIntElt -> GrpAb, Map
pQuotient(G, p, c) : GrpMat, RngIntElt, RngIntElt -> GrpPC, Map, SeqEnum, BoolElt
NilpotentQuotient(G, c) : GrpMat, RngIntElt -> GrpGPC, Map
SolvableQuotient(G): GrpMat -> GrpPC, Map
PCGroup(G): GrpMat -> GrpPC, Map
Example GrpMatGen_SpecialQuotient (H59E16)
Orbits and Stabilizers
u * g : ModTupRngElt, GrpMatElt -> ModTupRngElt
y ^ g : Elt, GrpMatElt -> Elt
y ^ G : Elt, GrpMat -> SetEnum
OrbitBounded(G, y, b) : GrpMat, Elt, RngIntElt -> BoolElt, SetEnum
Orbits(G) : GrpMat -> [ SetIndx ]
LineOrbits(G) : GrpMat -> [ SetIndx ]
OrbitClosure(G, S) : GrpMat, { Elt } -> GSet
Stabilizer(G, y) : GrpMat, Elt -> GrpMat
Example GrpMatGen_Orbits (H59E17)
Orbit and Stabilizer Functions for Large Groups
OrbitsOfSpaces(G, k) : GrpMat, RngIntElt -> SeqEnum
NumberOfFixedSpaces(x, s) : GrpMatElt, RngIntElt -> RngIntElt
Example GrpMatGen_OrbitsOfSpaces (H59E18)
EstimateOrbit(G, v: parameters) : GrpMat, ModTupFldElt -> RngIntElt, RngIntElt, RngIntElt
ApproximateStabiliser(G, A, U: parameters) : GrpMat, GrpMat, ModTupFld -> GrpMat, GrpMat, RngIntElt, RngIntElt, RngIntElt
Example GrpMatGen_OrbitsOfSpaces (H59E19)
StabiliserOfSpaces(Q) : SeqEnum -> GrpMat, SeqEnum
Example GrpMatGen_StabiliserOfSpaces (H59E20)
IsUnipotent(G) : GrpMat -> BoolElt
UnipotentStabiliser(G, U: parameters) : GrpMat, ModTupFld -> GrpMat, ModTupFld, GrpMatElt, GrpSLPElt
Example GrpMatGen_UnipotentStabiliser (H59E21)
Action on Orbits
OrbitAction(G, T) : GrpMat, Elt -> Hom(Grp), GrpPerm, GrpMat
OrbitActionBounded(G, T, b) : GrpMat, Elt, RngIntElt -> BoolElt, Hom(Grp), GrpPerm, GrpMat
OrbitImage(G, T) : GrpMat, Set -> GrpPerm
OrbitImageBounded(G, T, b) : GrpMat, Set, RngIntElt -> BoolElt, GrpPerm
OrbitKernel(G, T) : GrpMat, Set -> GrpMat
OrbitKernelBounded(G, T, b) : GrpMat, Set, RngIntElt -> BoolElt, GrpMat
Example GrpMatGen_Actions (H59E22)
Action on a Coset Space
CosetAction(G, H) : GrpMat, GrpMat -> Hom(Grp), GrpPerm, GrpMat
CosetImage(G, H) : GrpMat, GrpMat -> GrpPerm
CosetKernel(G, H) : GrpMat, GrpMat -> GrpMat
Example GrpMatGen_CosetAction (H59E23)
Action on the Natural G-Module
GModule(G) : GrpMat -> ModGrp
IsIrreducible(G) : GrpMat -> BoolElt, ModGrp
SubmoduleAction(G, S) : GrpMat -> Map, GrpMat
SubmoduleImage(G, S) : GrpMat -> GrpMat
QuotientModuleAction(G, S) : GrpMat -> Map, GrpMat
QuotientModuleImage(G, S) : GrpMat -> GrpMat
IsAbsolutelyIrreducible(G) : GrpMat -> BoolElt
AbsoluteRepresentation(G) : GrpMat -> GrpMat, Map
MinimalField(G) : GrpMat -> FldFin
Normal and Subnormal Subgroups
Characteristic Subgroups and Subgroup Series
Centre(G) : GrpMat -> GrpMat
DerivedLength(G) : GrpMat -> RngIntElt
DerivedSeries(G) : GrpMat -> [ GrpMat ]
CommutatorSubgroup(G) : GrpMat -> GrpMat
[Future release] FittingSubgroup(G) : GrpMat -> GrpMat
LowerCentralSeries(G) : GrpMat -> [ GrpMat ]
NilpotencyClass(G) : GrpMat -> RngIntElt
H ^ G : GrpMat -> GrpMat
SolubleResidual(G) : GrpMat -> GrpMat
SubnormalSeries(G, H) : GrpMat, GrpMat -> [ GrpMat ]
UpperCentralSeries(G) : GrpMat -> [ GrpMat ]
Example GrpMatGen_Series (H59E24)
The Soluble Radical and its Quotient
Radical(G) : GrpMat -> GrpMat
RadicalQuotient(G) : GrpMat -> GrpPerm, Hom(Grp), GrpMat
ElementaryAbelianSeries(G: parameters) : GrpMat -> [ GrpMat ]
ElementaryAbelianSeriesCanonical(G) : GrpMat -> [ GrpMat ]
Composition and Chief Factors
CompositionFactors(G) : : GrpMat -> [ <RngIntElt, RngIntElt, RngIntElt> ]
ChiefFactors(G) : GrpMat -> [ <RngIntElt, RngIntElt, RngIntElt, RngIntElt> ]
ChiefSeries(G) : GrpMat -> [ GrpMat ], [ <RngIntElt, RngIntElt, RngIntElt, RngIntElt> ]
Example GrpMatGen_CompositionFactors (H59E25)
Coset Tables and Transversals
CosetTable(G, H) : Grp, Grp -> Hom(Grp)
Transversal(G, H) : GrpMat, GrpMat -> {@ GrpMatElt @}, Map
Presentations
FPGroup(G) : GrpMat :-> GrpFP, Hom(Grp)
FPGroupStrong(G) : GrpMat :-> GrpFP, Hom(Grp)
Matrices as Words
WordGroup(G) : GrpMat -> GrpSLP, Map
InverseWordMap(G) : GrpMat -> Map
Automorphism Groups
AutomorphismGroup(G: parameters) : GrpMat -> GrpAuto
Example GrpMatGen_Automorphisms (H59E26)
IsIsomorphic(G, H: parameters) : GrpMat, GrpMat -> BoolElt, Hom(Grp)
Example GrpMatGen_Isomorphism (H59E27)
Representation Theory
LinearCharacters(G) : GrpMat -> [ Chtr ]
CharacterTable(G: parameters) : GrpMat -> TabChtr
PermutationCharacter(G, H) : GrpMat, GrpMat -> AlgChtrElt
GModule(G) : GrpMat -> ModGrp
GModule(G, A) : GrpMat, AlgMat -> ModGrp
GModule(G, Q) : GrpMat, [ AlgMatElt ] -> ModGrp
GModule(G, A, B) : GrpMat, GrpMat, GrpMat -> ModGrp, Map
PermutationModule(G, H, R) : GrpMat, GrpMat, Rng -> ModGrp
ChangeOfBasisMatrix(G, S) : GrpMat, ModGrp -> AlgMatElt
Example GrpMatGen_GModule (H59E28)
Base and Strong Generating Set
Controlling Selection of a Base
GoodBasePoints(G: parameters) : GrpMat -> []
AssertAttribute(G, "Base", B) : GrpMat, MonStgElt, Tup ->
HasAttribute(G, "Base") : GrpMat, MonStgElt -> BoolElt, Tup
AssertAttribute(GrpMat, "FirstBasicOrbitBound", n) : Cat, MonStgElt, RngIntElt ->
HasAttribute(GrpMat, "FirstBasicOrbitBound") : Cat, MonStgElt -> BoolElt, RngIntElt
Construction of a Base and Strong Generating Set
BSGS(G) : GrpMat ->
RandomSchreier(G: parameters) : GrpMat ->
ToddCoxeterSchreier(G) : GrpMat : ->
Verify(G) : GrpMat ->
Defining Values for Attributes
AssertAttribute(G, "Order", n) : GrpMat, MonStgElt, RngIntElt ->
AssertAttribute(G, "IsVerified", b) : GrpMat, MonStgElt, BoolElt ->
HasAttribute(G, "Order") : GrpMat, MonStgElt -> RngIntElt
HasAttribute(G, "IsVerified") : GrpMat, MonStgElt -> BoolElt
Accessing the Base and Strong Generating Set
Base(G) : GrpMat -> [Elt]
BasePoint(G, i) : GrpMat, RngIntElt -> Elt
BasicOrbit(G, i) : GrpMat, RngIntElt -> SetIndx
BasicOrbitLength(G, i) : GrpMat, RngIntElt -> RngIntElt
BasicOrbitLengths(G) : GrpMat -> [RngIntElt]
BasicStabilizer(G, i) : GrpMat, RngIntElt -> GrpMat
BasicStabilizerChain(G) : GrpMat -> [GrpMat]
NumberOfStrongGenerators(G) : GrpMat -> RngIntElt
StrongGenerators(G) : GrpMat -> SetIndx(GrpMat)
Conversion to a PC-Group
PolycyclicGenerators(G) : GrpMat -> [ GrpPCElt ]
PCGroup(G) : GrpMat -> GrpPC, Map
Soluble Group Functions
pCentralSeries(G, p) : GrpMat, RngIntElt -> [ GrpMat ]
p-group Functions
IsSpecial(G) : GrpMat -> BoolElt
IsExtraSpecial(G) : GrpMat -> BoolElt
FrattiniSubgroup(G) : GrpMat -> GrpMat
JenningsSeries(G) : GrpMat -> [ GrpMat ]
Abelian Group Functions
AbelianInvariants(G) : GrpMat -> [ RngIntElt ]
Bibliography
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Version: V2.19 of
Mon Dec 17 14:40:36 EST 2012