MATRIX GROUPS OVER GENERAL RINGS
Acknowledgements
Introduction
Introduction to Matrix Groups
The Support
The Category of Matrix Groups
The Construction of a Matrix Group
Creation of a Matrix Group
Construction of the General Linear Group
Construction of a Matrix Group Element
Construction of a General Matrix Group
Changing Rings
Coercion between Matrix Structures
Accessing Associated Structures
Homomorphisms
Construction of Extensions
Operations on Matrices
Arithmetic with Matrices
Predicates for Matrices
Matrix Invariants
Global Properties
Group Order
Membership and Equality
Set Operations
Abstract Group Predicates
Conjugacy
Conjugacy in Classical Groups
Subgroups
Construction of Subgroups
Elementary Properties of Subgroups
Standard Subgroups
Low Index Subgroups
Conjugacy Classes of Subgroups
Quotient Groups
Construction of Quotient Groups
Abelian, Nilpotent and Soluble Quotients
Matrix Group Actions
Orbits and Stabilizers
Orbit and Stabilizer Functions for Large Groups
Action on Orbits
Action on a Coset Space
Action on the Natural G-Module
Normal and Subnormal Subgroups
Characteristic Subgroups and Subgroup Series
The Soluble Radical and its Quotient
Composition and Chief Factors
Coset Tables and Transversals
Presentations
Presentations
Matrices as Words
Automorphism Groups
Representation Theory
Base and Strong Generating Set
Introduction
Controlling Selection of a Base
Construction of a Base and Strong Generating Set
Defining Values for Attributes
Accessing the Base and Strong Generating Set
Soluble Matrix Groups
Conversion to a PC-Group
Soluble Group Functions
p-group Functions
Abelian Group Functions
Bibliography
Introduction
Introduction to Matrix Groups
The Support
The Category of Matrix Groups
The Construction of a Matrix Group
Creation of a Matrix Group
Construction of the General Linear Group
GeneralLinearGroup(n, R) : RngIntElt, Rng -> GrpMat
Example GrpMatGen_Create (H66E1)
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 (H66E2)
Construction of a General Matrix Group
MatrixGroup< n, R | L > : RngIntElt, Rng, List -> GrpMat
Example GrpMatGen_Constructor (H66E3)
Example GrpMatGen_GLSylow (H66E4)
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
ProjectionLocalization(g, pi) : GrpMatElt, Map -> 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 (H66E5)
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 (H66E6)
Operations on Matrices
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 (H66E7)
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 (H66E8)
Global Properties
Group Order
IsFinite(G) : GrpMat -> Bool, RngIntElt
Order(G) : GrpMat -> RngIntElt
FactoredOrder(G) : GrpMat -> [ <RngIntElt, RngIntElt> ]
Exponent(G) : GrpMat -> RngIntElt
Example GrpMatGen_Order (H66E9)
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 (H66E10)
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 (H66E11)
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
ClassRepresentativeFromInvariants(G, p, h, t) : GrpMat, SeqEnum, SeqEnum, FldFinElt -> GrpMatElt
IsConjugate(G, g, h) : GrpMat, GrpMatElt, GrpMatElt -> BoolElt, GrpMatElt | Unass
NumberOfClasses(G) : GrpMat -> RngIntElt
PowerMap(G) : GrpMat -> Map
AssertAttribute(G, "Classes", Q) : GrpMat, MonStgElt, SeqEnum ->
Example GrpMatGen_RationalMatrixGroupDatabase (H66E12)
Conjugacy in Classical Groups
DualPolynomial(f) : RngUPolElt -> RngUPolElt
StarIrreduciblePolynomials(F,d) : FldFin, RngIntElt -> SeqEnum
PhiDual(f,phi) : RngUPolElt -> RngUPolElt
PhiIrreduciblePolynomials(F,d) : FldFin, RngIntElt -> SeqEnum[Tup]
ExtendedSymplecticGroup(n,q,m) : RngIntElt, RngIntElt, RngIntElt -> GrpMat
IndexOfSp(G) : GrpMat -> RngIntElt
TildeDualPolynomial(f) : RngUPolElt -> RngUPolElt
TildeIrreduciblePolynomials(q,d) : RngIntElt, RngIntElt -> SeqEnum
ExtendedSpecialUnitaryGroup(n,q,m) : RngIntElt, RngIntElt, RngIntElt -> GrpMat
IndexOfSU(G) : GrpMat -> RngIntElt
ClassicalConjugacyClasses(G) : GrpMat -> SeqEnum, SetIndx
ClassicalConjugacyClasses(type,d,q) : MonStgElt, RngIntElt, RngIntElt -> SeqEnum, SetIndx
ClassicalCentralizer(G,g) : GrpMat, GrpMatElt -> GrpMat
ClassicalCentraliserOrder(G,g) : GrpMat, GrpMatElt -> RngIntEltFact
ClassicalClassSize(G,g) : GrpMat, GrpMatElt -> RngIntElt
ClassicalIsConjugate(G,g,h) : GrpMat, GrpMatElt, GrpMatElt -> BoolElt, GrpMatElt
Example GrpMatGen_Class-calculations-I (H66E13)
ClassicalClassMap(G) : GrpMat -> Map
ClassesForFixedSemisimple(G,x) : GrpMat, GrpMatElt -> SeqEnum, SetIndx
IsometryGroupClassLabel(type, g) : MonStgElt, GrpMatElt -> SetMulti
Example GrpMatGen_Class-calculations (H66E14)
Example GrpMatGen_Class-calculations-III (H66E15)
Example GrpMatGen_Invlayer (H66E16)
UnipotentClasses(type,d,q) : MonStgElt, RngIntElt, RngIntElt -> SeqEnum, SeqEnum
SemisimpleClasses(type,d,q) : MonStgElt, RngIntElt, RngIntElt -> SeqEnum
IsometryGroupNumberOfClasses(type, n): MonStgElt, RngIntElt -> RngUPolElt
Example GrpMatGen_Class-calculations-IV (H66E17)
ProjectiveClassicalClasses(type,d,q) : MonStgElt, RngIntElt, RngIntElt -> SeqEnum, GrpPerm, HomGrp, SeqEnum
ProjectiveClassicalCentraliser(G, g) : GrpMat, GrpMatElt -> GrpMat
ProjectiveClassicalIsConjugate(G, g, h) : GrpMat, GrpMatElt, GrpMatElt-> BoolElt, GrpMatElt
Example GrpMatGen_ProjectiveClasses (H66E18)
Example GrpMatGen_ProjWithMatrices (H66E19)
SpinConjugacyClasses(G) : GrpMat -> SeqEnum, SeqEnum
SpinCentralizer(G,g) : GrpMat, GrpMatElt -> GrpMat
SpinIsConjugate(G,g,h) : GrpMat, GrpMatElt, GrpMatElt -> BoolElt, GrpMatElt
SpinClassMap(G) : GrpMat -> Map
Subgroups
Construction of Subgroups
sub<G | L> : GrpMat, List -> GrpMat
ncl<G | L> : GrpMat, List -> GrpMat
Example GrpMatGen_Subgroups (H66E20)
Elementary Properties of Subgroups
Index(G, H) : GrpMat, GrpMat -> RngIntElt
FactoredIndex(G, H) : GrpMat, GrpMat -> [ <RngIntElt, RngIntElt> ]
IsCentral(G, H) : GrpMat, 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
Centraliser(G, g) : GrpMat, GrpMatElt -> GrpMat
Centraliser(G, H) : GrpMat, GrpMat -> GrpMat
Core(G, H) : GrpMat, GrpMat -> GrpMat
H ^ G : GrpMat, GrpMat -> GrpMat
Normalizer(G, H) : GrpMat, GrpMat -> GrpMat
GLNormalizer(H : parameter) : GrpMat -> GrpMat
SylowSubgroup(G, p) : GrpMat, RngIntElt -> GrpMat
pCore(G, p) : GrpMat, RngIntElt -> GrpMat
Low Index Subgroups
LowIndexSubgroups(G,n: parameters) : GrpMat, RngIntElt -> SeqEnum
LowIndexSubgroups(G, N, n: parameters) : GrpMat, RngIntElt -> SeqEnum
LowIndexSubgroupsCT(G, R : parameters) : GrpMat, RngIntElt -> [ GrpMat ]
Example GrpMatGen_LowIndexMatrixGroup (H66E21)
Conjugacy Classes of Subgroups
SubgroupClasses(G: parameters) : GrpMat -> [ rec< GrpMat, RngIntElt, RngIntElt, GrpFP> ]
MaximalSubgroups(G: parameters) : GrpMat -> [ rec< GrpMat, RngIntElt, RngIntElt, GrpFP> ]
MaximalSubgroups(G,N: parameters) : GrpMat, GrpMat -> [ rec< GrpMat, RngIntElt, RngIntElt, GrpFP> ]
SubgroupsLift(G, A, B, Q: parameters) : GrpMat, GrpMat, GrpMat, SeqEnum -> SeqEnum
IsConjugate(G, H, K) : GrpMat, GrpMat, GrpMat -> BoolElt, GrpMatElt | Unass
IsGLConjugate(H, K) : GrpMat, GrpMat -> BoolElt, GrpMatElt | Unass
Quotient Groups
Construction of Quotient Groups
quo<G | L> : GrpMat, List -> GrpPerm, Map
G / N : GrpMat, GrpMat -> GrpPerm
Example GrpMatGen_Quotient (H66E22)
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 (H66E23)
Matrix Group Actions
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 (H66E24)
Orbit and Stabilizer Functions for Large Groups
OrbitsOfSpaces(G, k) : GrpMat, RngIntElt -> SeqEnum
NumberOfFixedSpaces(x, s) : GrpMatElt, RngIntElt -> RngIntElt
Example GrpMatGen_OrbitsOfSpaces (H66E25)
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 (H66E26)
StabiliserOfSpaces(Q) : SeqEnum -> GrpMat, SeqEnum
Example GrpMatGen_StabiliserOfSpaces (H66E27)
IsUnipotent(G) : GrpMat -> BoolElt
UnipotentStabiliser(G, U: parameters) : GrpMat, ModTupFld -> GrpMat, ModTupFld, GrpMatElt, GrpSLPElt
Example GrpMatGen_UnipotentStabiliser (H66E28)
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, SetIndx
OrbitImageBounded(G, T, b) : GrpMat, Set, RngIntElt -> BoolElt, GrpPerm, SetIndx
OrbitKernel(G, T) : GrpMat, Set -> GrpMat
OrbitKernelBounded(G, T, b) : GrpMat, Set, RngIntElt -> BoolElt, GrpMat
Example GrpMatGen_Actions (H66E29)
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 (H66E30)
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 (H66E31)
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 (H66E32)
Coset Tables and Transversals
CosetTable(G, H) : Grp, Grp -> Hom(Grp)
Transversal(G, H) : GrpMat, GrpMat -> {@ GrpMatElt @}, Map
Presentations
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 (H66E33)
IsIsomorphic(G, H: parameters) : GrpMat, GrpMat -> BoolElt, Hom(Grp)
Example GrpMatGen_Isomorphism (H66E34)
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 (H66E35)
Base and Strong Generating Set
Introduction
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 ->
RandomSchreierBounded(G, L: parameters) : GrpMat, RngIntElt -> BoolElt
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)
Soluble Matrix Groups
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
PrimaryAbelianInvariants(G) : GrpMat -> [ RngIntElt ]
PrimaryAbelianBasis(G) : GrpMat -> [ GrpMatElt ], [ RngIntElt ]
Bibliography
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