PERMUTATION GROUPS
Acknowledgements
Introduction
Terminology
The Category of Permutation Groups
The Construction of a Permutation Group
Creation of a Permutation Group
Construction of the Symmetric Group
Construction of a Permutation
Construction of a General Permutation Group
Elementary Properties of a Group
Accessing Group Information
Group Order
Abstract Properties of a Group
Homomorphisms
Building Permutation Groups
Some Standard Permutation Groups
Direct Products and Wreath Products
Permutations
Coercion
Arithmetic with Permutations
Properties of Permutations
Predicates for Permutations
Set Operations
Conjugacy
Subgroups
Construction of a Subgroup
Membership and Equality
Elementary Properties of a Subgroup
Standard Subgroups
Maximal Subgroups
Conjugacy Classes of Subgroups
Classes of Subgroups Satisfying a Condition
Quotient Groups
Construction of Quotient Groups
Abelian, Nilpotent and Soluble Quotients
Permutation Group Actions
G-Sets
Creating a G-Set
Images, Orbits and Stabilizers
Action on a G-Space
Action on Orbits
Action on a G-invariant Partition
Action on a Coset Space
Reduced Permutation Actions
The Jellyfish Algorithm
Normal and Subnormal Subgroups
Characteristic Subgroups and Normal Series
Maximal and Minimal Normal Subgroups
Lattice of Normal Subgroups
Composition and Chief Series
The Socle
The Soluble Radical and its Quotient
Complements and Supplements
Abelian Normal Subgroups
Cosets and Transversals
Cosets
Transversals
Presentations
Generators and Relations
Permutations as Words
Automorphism Groups
Cohomology
Representation Theory
Identification
Identification as an Abstract Group
Identification as a Permutation Group
Base and Strong Generating Set
Construction of a Base and Strong Generating Set
Defining Values for Attributes
Accessing the Base and Strong Generating Set
Working with a Base and Strong Generating Set
Modifying a Base and Strong Generating Set
Permutation Representations of Linear Groups
Permutation Group Databases
Ordering of Permutation Groups
Ordered Partition Stacks
Construction of Ordered Partition Stacks
Properties of Ordered Partition Stacks
Operations on Ordered Partition Stacks
Bibliography
Introduction
Terminology
The Category of Permutation Groups
The Construction of a Permutation Group
Creation of a Permutation Group
Construction of the Symmetric Group
Sym(n) : RngIntElt -> GrpPerm
Sym(X) : Set -> GrpPerm
StandardGroup(G) : GrpPerm -> GrpPerm, Map
Example GrpPerm_Sym (H65E1)
Construction of a Permutation
elt< G | L > : GrpPerm, List(Elt) -> GrpPermElt
G ! Q : GrpPerm, [ Elt ] -> GrpPermElt
G ! (...)(...)...(...) : GrpPerm, Cycles -> GrpPermElt
G ! \(...)(...)...(...) : GrpPerm, LiteralCycles -> GrpPermElt
G ! Q : GrpPerm, SeqEnum[SetIndx] -> GrpPermElt
ElementToSequence(g) : GrpPermElt -> [ Elt ]
Identity(G) : Grp -> GrpPermElt
Example GrpPerm_Permutations (H65E2)
Construction of a General Permutation Group
PermutationGroup< X | L > : Set, List -> GrpPerm
PermutationGroup< n | L > : RngIntElt, List -> GrpPerm
Example GrpPerm_Hessian (H65E3)
Elementary Properties of a Group
Accessing Group Information
G . i : GrpPerm, RngIntElt -> GrpPermElt
Degree(G) : GrpPermElt -> RngIntElt
Generators(G) : GrpPerm -> { GrpPermElt }
GeneratorsSequence(G) : GrpPerm -> [ GrpPermElt ]
NumberOfGenerators(G) : GrpPerm -> RngIntElt
FewGenerators(G) : GrpPerm -> [GrpPermElt]
Generic(G) : GrpPerm -> GrpPerm
Parent(g) : GrpPermElt -> GrpPerm
GSet(G) : GrpPerm -> GSet
Example GrpPerm_BasicAccess (H65E4)
Group Order
Order(G) : GrpPerm -> RngIntElt
FactoredOrder(G) : GrpPerm -> [ <RngIntElt, RngIntElt> ]
Abstract Properties of a Group
IsAbelian(G) : GrpPerm -> BoolElt
IsCyclic(G) : GrpPerm -> BoolElt
IsElementaryAbelian(G) : GrpPerm -> BoolElt
IsSpecial(G) : GrpPerm -> BoolElt
IsExtraSpecial(G) : GrpPerm -> BoolElt
IsNilpotent(G) : GrpPerm -> BoolElt
IsSoluble(G) : GrpPerm -> BoolElt
IsPerfect(G) : GrpPerm -> BoolElt
IsSimple(G) : GrpPerm -> BoolElt
IsWreathProduct(G) : GrpPerm -> BoolElt, GrpPerm, GrpPerm, GrpPerm
Example GrpPerm_BasicProperties (H65E5)
Homomorphisms
hom<G -> H | L> : GrpPerm, List -> Map
Domain(f) : Map -> Grp
Codomain(f) : Map -> Grp
Image(f) : Map -> Grp
Kernel(f) : Map -> Grp
IsHomomorphism(G, H, Q) : GrpPerm, GrpPerm, SeqEnum[GrpPermElt] -> Bool, Map
Example GrpPerm_Homomorphism (H65E6)
Building Permutation Groups
Some Standard Permutation Groups
AbelianGroup(GrpPerm, Q) : Cat, [ RngIntElt ] -> GrpPerm
AlternatingGroup(GrpPerm, n) : Cat, RngIntElt -> GrpPerm
CyclicGroup(GrpPerm, n) : Cat, RngIntElt -> GrpPerm
DihedralGroup(GrpPerm, n) : Cat, RngIntElt -> GrpPerm
Sym(GrpPerm, n) : Cat, RngIntElt -> GrpPerm
ExtraSpecialGroup(GrpPerm, p, n : parameters) : Cat, RngIntElt, RngIntElt -> GrpPerm
YoungSubgroup(L) : [RngIntElt] -> GrpPerm
Example GrpPerm_StandardGroups (H65E7)
Direct Products and Wreath Products
DirectProduct(G, H) : GrpPerm, GrpPerm -> GrpPerm, [ Hom(Grp) ], [ Hom(Grp) ]
DirectProduct(Q) : [ GrpPerm ] -> GrpPerm, [ Hom(Grp) ], [ Hom(Grp) ]
PrimitiveWreathProduct(G, H) : GrpPerm, GrpPerm -> GrpPerm
PrimitiveWreathProduct(Q) : [ GrpPerm ] -> GrpPerm
WreathProduct(G, H) : GrpPerm, GrpPerm -> GrpPerm, SeqEnum[Map], Map, Map
WreathProduct(Q) : [ GrpPerm ] -> GrpPerm
WreathProduct(B) : GSet -> GrpPerm, GrpPerm, GrpPerm
WreathProduct(G, B) : GrpPerm, GSet -> GrpPerm, GrpPerm, GrpPerm
Example GrpPerm_Products (H65E8)
Permutations
Coercion
G ! g : GrpPerm, GrpPermElt -> GrpPermElt
G !! H : GrpPerm, GrpPerm -> GrpPerm
Arithmetic with Permutations
g * h : GrpPermElt, GrpPermElt -> GrpPermElt
g ^ n : GrpPermElt, RngIntElt -> GrpPermElt
g / h : GrpPermElt, GrpPermElt -> GrpPermElt
g ^ h : GrpPermElt, GrpPermElt -> GrpPermElt
(g, h) : GrpPermElt, GrpPermElt -> GrpPermElt
(g1, ..., gr) : GrpPermElt, ..., GrpPermElt -> GrpPermElt
Properties of Permutations
CycleStructure(g) : GrpPermElt -> [ <RngIntElt, RngIntElt> ]
Degree(g) : GrpPermElt -> RngIntElt
IsEven(g) : GrpPermElt -> BoolElt
Sign(g) : GrpPermElt -> RngIntElt
Order(g) : GrpPermElt -> RngIntElt
Predicates for Permutations
g eq h : GrpPermElt, GrpPermElt -> BoolElt
g ne h : GrpPermElt, GrpPermElt -> BoolElt
IsId(g) : GrpPermElt -> BoolElt
Example GrpPerm_Arithmetic (H65E9)
Set Operations
G * H : GrpPerm, GrpPerm -> { GrpPermElt }
ElementSet(G, H) : GrpPerm, GrpPerm -> { GrpPermElt }
NumberingMap(G) : GrpPerm -> Map
RandomProcess(G) : GrpPerm -> Process
Random(G: parameters) : GrpPerm -> GrpPermElt
Random(P) : Process -> GrpPermElt
Representative(G) : GrpPerm -> GrpPermElt
Example GrpPerm_SetOperations (H65E10)
Example GrpPerm_SetOperations-2 (H65E11)
Conjugacy
Class(H, x) : GrpPerm, GrpPermElt -> { GrpPermElt }
ConjugacyClasses(G: parameters) : GrpPerm -> [ <RngIntElt, RngIntElt, GrpPermElt> ]
ClassRepresentative(G, x) : GrpPerm, GrpPermElt -> GrpPermElt
ClassCentraliser(G, i) : GrpPerm, RngIntElt -> GrpPerm
ClassMap(G: parameters) : GrpPerm -> Map
IsConjugate(G, g, h: parameters) : GrpPerm, GrpPermElt, GrpPermElt -> BoolElt, GrpPermElt
IsConjugate(G, H, K: parameters) : GrpPerm, GrpPerm, GrpPerm -> BoolElt, GrpPermElt
Exponent(G) : GrpPerm -> RngIntElt
NumberOfClasses(G) : GrpPerm -> RngIntElt
PowerMap(G) : GrpPerm -> Map
AssertAttribute(G, "Classes", Q) : GrpPerm, MonStgElt, SeqEnum ->
Example GrpPerm_Classes (H65E12)
Example GrpPerm_Classes-2 (H65E13)
Subgroups
Construction of a Subgroup
sub<G | L> : GrpPerm, List -> GrpPerm
ncl<G | L> : GrpPerm, List -> GrpPerm
Example GrpPerm_Constructors (H65E14)
Example GrpPerm_Constructors-2 (H65E15)
Example GrpPerm_Constructors-3 (H65E16)
Membership and Equality
g in G : GrpPermElt, GrpPerm -> BoolElt
g notin G : GrpPermElt, GrpPerm -> BoolElt
S subset G : { GrpPermElt }, GrpPerm -> BoolElt
S notsubset G : { GrpPermElt }, GrpPerm -> BoolElt
H subset G : GrpPerm, GrpPerm -> BoolElt
H notsubset G : GrpPerm, GrpPerm -> BoolElt
H eq G : GrpPerm, GrpPerm -> BoolElt
H ne G : GrpPerm, GrpPerm -> BoolElt
Elementary Properties of a Subgroup
Index(G, H) : GrpPerm, GrpPerm -> RngIntElt
FactoredIndex(G, H) : GrpPerm, GrpPerm -> [ <RngIntElt, RngIntElt> ]
IsCentral(G, H) : GrpPerm, GrpPerm -> BoolElt
IsNormal(G, H) : GrpPerm, GrpPerm -> BoolElt
IsSelfNormalizing(G, H) : GrpPerm, GrpPerm -> BoolElt
IsSubnormal(G, H) : GrpPerm, GrpPerm -> BoolElt
Standard Subgroups
H ^ g : GrpPerm, GrpPermElt -> GrpPerm
H meet K : GrpPerm, GrpPerm -> GrpPerm
IntersectionWithNormalSubgroup(G, N: parameters) : GrpPerm, GrpPerm -> GrpPerm
CommutatorSubgroup(G, H, K) : GrpPerm, GrpPerm, GrpPerm -> GrpPerm
Centralizer(G, g: parameters) : GrpPerm, GrpPermElt -> GrpPerm
Centralizer(G, H) : GrpPerm, GrpPerm -> GrpPerm
CentralizerOfNormalSubgroup(G, H) : GrpPerm, GrpPerm -> GrpPerm
SectionCentraliser(G, H, K) : GrpPerm, GrpPerm, GrpPerm -> GrpPerm
Core(G, H) : GrpPerm, GrpPerm -> GrpPerm
H ^ G : GrpPerm, GrpPerm -> GrpPerm
Normalizer(G, H: parameters) : GrpPerm, GrpPerm -> GrpPerm
SymmetricNormalizer(G) : GrpPerm -> GrpPerm
SylowSubgroup(G, p) : GrpPerm, RngIntElt -> GrpPerm
Example GrpPerm_SubgroupConstructions (H65E17)
Maximal Subgroups
IsMaximal(G, H: parameters) : GrpPerm, GrpPerm -> BoolElt
IsProbablyMaximal(G, H: parameters) : GrpPerm, GrpPerm -> BoolElt
MaximalSubgroups(G: parameters) : GrpPerm -> [ rec< GrpPerm, RngIntElt, RngIntElt, GrpFP> ]
Example GrpPerm_Maximals (H65E18)
MaximalSubgroups(G,N: parameters) : GrpPerm, GrpPerm -> [ rec< GrpPerm, RngIntElt, RngIntElt, GrpFP> ]
Conjugacy Classes of Subgroups
SubgroupClasses(G: parameters) : GrpPerm -> [ rec< GrpPerm, RngIntElt, RngIntElt, GrpFP> ]
SubgroupsLift(G, A, B, Q: parameters) : GrpPerm, GrpPerm, GrpPerm, SeqEnum -> SeqEnum
LowIndexSubgroups(G, n: parameters) : GrpPerm, RngIntElt -> SeqEnum
LowIndexSubgroups(G, N, n: parameters) : GrpPerm, RngIntElt -> SeqEnum
Example GrpPerm_Subgroups (H65E19)
Example GrpPerm_low-index-subs (H65E20)
Example GrpPerm_Subgroups-2 (H65E21)
SubgroupLattice(G) : GrpPerm -> SubGrpLat
BurnsideMatrix(G) : GrpPerm -> AlgMatElt
DisplayBurnsideMatrix(G) : GrpPerm ->
TableOfMarks(G) : GrpPerm -> AlgMatElt
Classes of Subgroups Satisfying a Condition
NormalSubgroups(G: parameters) : GrpPerm -> [ rec< GrpPerm, RngIntElt, RngIntElt, GrpFP> ]
ElementaryAbelianSubgroups(G: parameters) : GrpPerm -> [ rec< GrpPerm, RngIntElt, RngIntElt, GrpFP> ]
CyclicSubgroups(G: parameters) : GrpPerm -> [ rec< GrpPerm, RngIntElt, RngIntElt, GrpFP> ]
AbelianSubgroups(G: parameters) : GrpPerm -> [ rec< GrpPerm, RngIntElt, RngIntElt, GrpFP> ]
NilpotentSubgroups(G: parameters) : GrpPerm -> [ rec< GrpPerm, RngIntElt, RngIntElt, GrpFP> ]
SolvableSubgroups(G: parameters) : GrpPerm -> [ rec< GrpPerm, RngIntElt, RngIntElt, GrpFP> ]
PerfectSubgroups(G: parameters) : GrpPerm -> [ rec< GrpPerm, RngIntElt, RngIntElt, GrpFP> ]
NonsolvableSubgroups(G: parameters) : GrpPerm -> [ rec< GrpPerm, RngIntElt, RngIntElt, GrpFP> ]
SimpleSubgroups(G: parameters) : GrpPerm -> [ rec< GrpPerm, RngIntElt, RngIntElt, GrpFP> ]
Quotient Groups
Construction of Quotient Groups
quo<G | L> : GrpPerm, List -> GrpPerm, Map
G / N : GrpPerm, GrpPerm -> GrpPerm
Example GrpPerm_Quotient (H65E22)
Abelian, Nilpotent and Soluble Quotients
AbelianQuotient(G) : GrpPerm -> GrpAb, Map
ElementaryAbelianQuotient(G, p) : GrpPerm, RngIntElt -> GrpAb, Map
pQuotient(G, p, c) : GrpPerm, RngIntElt, RngIntElt -> GrpPC, Map, SeqEnum, BoolElt
NilpotentQuotient(G, c) : GrpPerm, RngIntElt -> GrpGPC, Map
SolvableQuotient(G): GrpPerm -> GrpPC, Map, SeqEnum, MonStgElt
Example GrpPerm_SpecialQuotient (H65E23)
Permutation Group Actions
G-Sets
Creating a G-Set
GSetFromIndexed(G, Y) : GrpPerm, SetIndx -> GSet
GSet(G, X, Y) : GrpPerm, GSet, SetEnum -> GSet
GSet(G) : GrpPerm -> GSet
GSet(G, Y, f) : GrpPerm, Set, Map -> GSet
Action(Y) : GSet -> Map
Group(Y) : GSet -> GrpPerm
Labelling(G) : GrpPerm -> SetIndx
Degree(g, Y) : GrpPermElt, GSet -> RngIntElt
Degree(G, Y) : GrpPerm, GSet -> RngIntElt
Support(g, Y) : GrpPermElt, GSet -> { Elt }
Support(G, Y) : GrpPerm, GSet -> { Elt }
Example GrpPerm_GSets (H65E24)
Images, Orbits and Stabilizers
x ^ g : Elt, GrpPermElt -> Elt
Image(g, Y, y) : GrpPermElt, GSet, Elt -> Elt
Fix(g, Y): GrpPermElt, GSet -> { Elt }
Fix(G, Y) : GrpPerm, GSet -> { Elt }
x ^ G : Elt, GrpPerm -> GSet
Cycle(e, x) : GrpPermElt, Elt -> SetIndx
CycleDecomposition(e) : GrpPermElt -> SeqEnum[SetIndx]
Orbit(G, Y, y) : GrpPerm, GSet, Elt -> GSet
Orbits(G, Y) : GrpPerm, GSet -> [ GSet ]
OrbitRepresentatives(G) : GrpPerm -> SeqEnum
OrbitClosure(G, Y, S) : GrpPerm, GSet, { Elt } -> GSet
IsConjugate(G, Y, y, z) : GrpPerm, GSet, Elt, Elt -> BoolElt, GrpPermElt
Stabilizer(G, Y, y) : GrpPerm, GSet, Elt -> GrpPerm
IsPrimitive(G, Y) : GrpPerm, GSet -> BoolElt
IsTransitive(G, Y) : GrpPerm, GSet -> BoolElt
IsTransitive(G, Y, k) : GrpPerm, GSet, RngIntElt -> BoolElt
IsSharplyTransitive(G, Y, k) : GrpPerm, GSet, RngIntElt -> BoolElt
Transitivity(G, Y) : GrpPerm, GSet -> RngIntElt
Homogeneity(G) : GrpPerm -> RngIntElt
IsRegular(G, Y) : GrpPerm, GSet -> BoolElt
IsSemiregular(G, Y) : GrpPerm, GSet -> BoolElt
IsSemiregular(G, Y, S) : GrpPerm, GSet, SetEnum -> BoolElt
IsFrobenius(G) : GrpPerm -> BoolElt
Example GrpPerm_Stabilizers (H65E25)
Action on a G-Space
Action(G, Y) : GrpPerm, GSet -> Hom(Grp), GrpPerm, GrpPerm
ActionImage(G, Y) : GrpPerm, GSet -> GrpPerm
ActionKernel(G, Y) : GrpPerm, GSet -> GrpPerm
IsFaithful(G, Y) : GrpPerm, GSet -> BoolElt
Example GrpPerm_Actions (H65E26)
Action on Orbits
OrbitAction(G, T) : GrpPerm, GSet -> Hom(Grp), GrpPerm, GrpPerm
OrbitImage(G, T) : GrpPerm, GSet -> GrpPerm
OrbitKernel(G, T) : GrpPerm, GSet -> GrpPerm
IsOrbit(G, S) : GrpPerm, { Elt } -> BoolElt
Example GrpPerm_OrbitActions (H65E27)
Action on a G-invariant Partition
IsBlock(G, S) : GrpPerm, { Elt } -> BoolElt
IsPrimitive(G) : GrpPerm -> BoolElt
MaximalPartition(G) : GrpPerm -> GSet
MinimalPartition(G: parameters) : GrpPerm -> GSet
MinimalPartitions(G: parameters) : GrpPerm -> [ GSet ]
MinimalBlocks(G: parameters) : GrpPerm -> [ SetEnum ]
AllPartitions(G) : GrpPerm -> SetEnum
BlocksAction(G, P) : GrpPerm, Any -> Hom(GrpPerm), GrpPerm, GrpPerm
BlocksImage(G, P) : GrpPerm, Any -> GrpPerm
BlocksKernel(G, P) : GrpPerm, Any -> GrpPerm
Example GrpPerm_BlocksActions (H65E28)
Example GrpPerm_BlocksActions-2 (H65E29)
Action on a Coset Space
CosetAction(G, H: parameters) : Grp, Grp -> Hom(Grp), GrpPerm, GrpPerm
CosetImage(G, H: parameters) : Grp, Grp -> GrpPerm
CosetKernel(G, H) : Grp, Grp -> Grp
Reduced Permutation Actions
TransitiveQuotient(G) : GrpPerm -> GrpPerm, Hom, GrpPerm
PrimitiveQuotient(G) : GrpPerm -> GrpPerm, Hom, GrpPerm
DegreeReduction(G) : GrpPerm -> GrpPerm, Hom
The Jellyfish Algorithm
JellyfishConstruction(G: parameters) : GrpPerm -> BoolElt
JellyfishImage(G) : GrpPerm -> GrpPerm
JellyfishImage(G, x: parameters) : GrpPerm, GrpPermElt -> GrpPermElt
JellyfishPreimage(G, x: parameters) : GrpPerm, GrpPermElt -> GrpPermElt
Normal and Subnormal Subgroups
Characteristic Subgroups and Normal Series
DerivedSeries(G) : GrpPerm -> [ GrpPerm ]
CompositionSeries(G) : GrpPerm -> [ GrpPerm ]
CommutatorSubgroup(G) : GrpPerm -> GrpPerm
SolubleResidual(G) : GrpPerm -> GrpPerm
DerivedLength(G) : GrpPerm -> RngIntElt
LowerCentralSeries(G) : GrpPerm -> [ GrpPerm ]
NilpotencyClass(G) : GrpPerm -> RngIntElt
UpperCentralSeries(G) : GrpPerm -> [ GrpPerm ]
Centre(G) : GrpPerm -> GrpPerm
Hypercentre(G) : GrpPerm -> GrpPerm
pCore(G, p) : GrpPerm, RngIntElt -> GrpPerm
pCoreQuotient(G, p) : GrpPerm, RngIntElt -> GrpPerm, Map, GrpPerm
FittingGroup(G) : GrpPerm -> GrpPerm
FrattiniSubgroup(G) : GrpPerm -> GrpPerm
JenningsSeries(G) : GrpPerm -> [ GrpPerm ]
pCentralSeries(G, p) : GrpPerm, RngIntElt -> [ GrpPerm ]
SubnormalSeries(G, H) : GrpPerm, GrpPerm -> [ GrpPerm ]
Example GrpPerm_Series (H65E30)
Maximal and Minimal Normal Subgroups
MaximalNormalSubgroup(G) : GrpPerm -> GrpPerm
MinimalNormalSubgroups(G) : GrpPerm -> [ GrpPerm ]
Lattice of Normal Subgroups
NormalSubgroups(G) : GrpPerm -> [ Rec ]
NormalLattice(G) : GrpPerm -> SubGrpLat
Example GrpPerm_NormalSubgroups (H65E31)
Composition and Chief Series
ChiefFactors(G) : GrpPerm -> [ <RngIntElt, RngIntElt, RngIntElt, RngIntElt> ]
ChiefSeries(G) : GrpPerm -> [ GrpPerm ], [ <RngIntElt, RngIntElt, RngIntElt, RngIntElt> ]
CompositionFactors(G) : GrpPerm -> [ <RngIntElt, RngIntElt, RngIntElt> ]
Example GrpPerm_CompFactors (H65E32)
PrimaryAbelianInvariants(G) : GrpPerm -> [ RngIntElt ]
PrimaryAbelianBasis(G) : GrpPerm -> [ GrpPermElt ], [ RngIntElt ]
The Socle
Socle(G) : GrpPerm -> GrpPerm
SocleFactor(G) : GrpPerm -> GrpPerm
SocleFactors(G) : GrpPerm -> [ GrpPerm ]
SocleSeries(G) : GrpPerm -> [ GrpPerm ]
EARNS(G) : GrpPerm -> GrpPerm
AffineGeneralLinearGroup(E) : GrpPerm -> GrpPerm
IsAffine(G) : GrpPerm -> BoolElt, GrpPerm
AffineAction(G) : GrpPerm -> Hom, GrpPerm, GrpPerm
AffineImage(G) : GrpPerm -> GrpPerm
AffineKernel(G) : GrpPerm -> GrpPerm
SocleAction(G) : GrpPerm -> Hom, GrpPerm, GrpPerm
SocleImage(G) : GrpPerm -> GrpPerm
SocleKernel(G) : GrpPerm -> GrpPerm
SocleQuotient(G) : GrpPerm -> GrpPerm, Hom, GrpPerm
RefineSection(G, M, N) : GrpPerm, GrpPerm, GrpPerm -> [ GrpPerm ]
Example GrpPerm_PrimitiveStructure (H65E33)
The Soluble Radical and its Quotient
Radical(G) : GrpPerm -> GrpPerm
RadicalQuotient(G) : GrpPerm -> GrpPerm, Hom(GrpPerm), GrpPerm
ElementaryAbelianSeries(G: parameters) : GrpPerm -> [ GrpPerm ]
ElementaryAbelianSeriesCanonical(G) : GrpPerm -> [ GrpPerm ]
Example GrpPerm_Radical (H65E34)
Complements and Supplements
Complements(G, M) : GrpPerm, GrpPerm -> [ GrpPerm ]
Complements(G, M, N) : GrpPerm, GrpPerm, GrpPerm -> [ GrpPerm ]
HasComplement(G, M) : GrpPerm, GrpPerm -> BoolElt, GrpPerm
Supplements(G, M) : GrpPerm, GrpPerm -> [ GrpPerm ]
Supplements(G, M, N) : GrpPerm, GrpPerm, GrpPerm -> [ GrpPerm ]
HasSupplement(G, M) : GrpPerm, GrpPerm -> BoolElt, GrpPerm
Example GrpPerm_Complements (H65E35)
Abelian Normal Subgroups
AbelianNormalSubgroup(G) : GrpPerm -> GrpPerm
AbelianNormalQuotient(G, H) : GrpPerm, GrpPerm -> GrpPerm, Hom(GrpPerm), GrpPerm
SolubleNormalQuotient(G, H) : GrpPerm -> GrpPerm, Hom(GrpPerm), GrpPerm
ElementaryAbelianNormalSubgroup(G) : GrpPerm -> GrpPerm
pElementaryAbelianNormalSubgroup(G, p) : GrpPerm, RngIntElt -> GrpPerm
MEANS(G) : GrpPerm -> GrpPerm
MEANS(G, N) : GrpPerm, GrpPerm -> GrpPerm
Cosets and Transversals
Cosets
H * g : GrpPerm, GrpPermElt -> Elt
DoubleCoset(G, H, g, K) : GrpPerm, GrpPerm, GrpPermElt, GrpPerm -> GrpPermDcosElt
DoubleCosetRepresentatives(G, H, K) : GrpPerm, GrpPerm, GrpPerm -> SeqEnum, SeqEnum
DoubleCosetCanonical(G, H, g, K: parameters) : GrpPerm, GrpPerm, GrpPermElt, GrpPerm -> SeqEnum, SeqEnum
ProcessLadder(L, G, U) : [GrpPerm], GrpPerm, GrpPerm -> Rec
GetRep(p, R) : GrpPermElt, Rec -> GrpPermElt
DeleteData(R) : Rec ->
YoungSubgroupLadder(L) : [RngIntElt] -> [GrpPerm]
StabilizerLadder(G, d) : GrpPerm, RngMPolElt -> [GrpPerm]
x in C : GrpPermElt, Elt -> BoolElt
x notin C : GrpPermElt, Elt -> BoolElt
C1 eq C2 : Elt, Elt -> BoolElt
C1 ne C2 : Elt, Elt -> BoolElt
# C : Elt -> RngIntElt
CosetTable(G, H) : Grp, Grp -> Map
[Future release] CosetTable(G, f) : Grp, Map -> Map
Transversals
Transversal(G, H) : GrpPerm, GrpPerm -> {@ GrpPermElt @} , Map
TransversalProcess(G, H) : GrpPerm, GrpPerm -> GrpPermTransProc
TransversalProcessRemaining(P) : GrpPermTransProc -> RngIntElt
TransversalProcessNext(P) : GrpPermTransProc -> GrpPermElt
ShortCosets(p, H, G) : GrpPermElt, GrpPerm, GrpPerm -> [GrpPermElt]
Presentations
Generators and Relations
FPGroup(G) : GrpPerm :-> GrpFP, Hom(Grp)
FPGroup(G, N) : GrpPerm, GrpPerm :-> GrpFP, Hom(Grp)
FPGroupStrong(G: parameters) : GrpPerm :-> GrpFP, Hom(Grp)
Permutations as Words
WordGroup(G) : GrpPerm -> GrpBB, Map
InverseWordMap(G) : GrpPerm -> Map
ActingWord(G, x, y) : GrpPerm, Elt, Elt -> GrpFPElt
Automorphism Groups
AutomorphismGroup(G: parameters) : GrpPerm -> GrpAuto
IsIsomorphic(G, H: parameters) : GrpPerm, GrpPerm -> BoolElt, Hom(Grp)
Example GrpPerm_Automorphisms (H65E36)
Cohomology
pMultiplicator(G, p) : GrpPerm, RngIntElt -> [ RngIntElt ]
pCover(G, F, p) : GrpPerm, GrpFP, RngIntElt -> GrpFP
CohomologicalDimension(G, M, i) : GrpPerm, ModRng, RngIntElt -> RngIntElt
ExtensionProcess(G, M, F) : GrpPerm, ModRng, GrpFP -> Process
Extension(P, Q) : Process -> GrpFP
SplitExtension(G, M, F) : GrpPerm, ModRng, GrpFP -> GrpFP
Example GrpPerm_Cohomology (H65E37)
Example GrpPerm_Cohomology-2 (H65E38)
Representation Theory
CharacterTable(G: parameters) : GrpPerm -> TabChtr
PermutationCharacter(G) : GrpPerm -> AlgChtrElt
PermutationCharacter(G, H) : GrpPerm, GrpPerm -> AlgChtrElt
GModule(G, S) : Grp, AlgMat -> ModGrp
GModule(G, A, B) : Grp, Grp, Grp -> ModGrp, Map
PermutationModule(G, H, R) : Grp, Grp, Rng -> ModGrp
PermutationModule(G, R) : GrpPerm, Rng -> ModGrp
Example GrpPerm_GModule (H65E39)
Identification
Identification as an Abstract Group
NameSimple(G) : GrpPerm -> <RngIntElt, RngIntElt, RngIntElt>
Identification as a Permutation Group
IsAlternating(G) : GrpPerm -> BoolElt
IsSymmetric(G) : GrpPerm -> BoolElt
IsAltsym(G : parameters) : GrpPerm -> BoolElt
TwoTransitiveGroupIdentification(G) : GrpPerm -> Tup
IsEven(G): GrpPerm -> BoolElt
RecogniseAlternatingOrSymmetric(G : parameters) : Grp, RngIntElt -> BoolElt, Map, Map, Map, Map, BoolElt
AlternatingOrSymmetricElementToWord(G, g): Grp, GrpElt -> BoolElt, GrpSLPElt
Example GrpPerm_RecogniseAltsym2 (H65E40)
RecogniseSymmetric(G, n: parameters) : Grp, RngIntElt -> BoolElt, Map, Map, Map, Map, BoolElt
SymmetricElementToWord (G, g) : Grp, GrpElt -> BoolElt, GrpSLPElt
RecogniseAlternating(G, n: parameters) : Grp, RngIntElt -> BoolElt, Map, Map, Map, Map, BoolElt
AlternatingElementToWord (G, g) : Grp, GrpElt -> BoolElt, GrpSLPElt
GuessAltsymDegree(G: parameters) : Grp -> BoolElt, MonStgElt, RngIntElt
Example GrpPerm_RecogniseAltsym2 (H65E41)
Base and Strong Generating Set
Construction of a Base and Strong Generating Set
BSGS(G) : GrpPerm ->
SimsSchreier(G: parameters) : GrpPerm ->
RandomSchreier(G: parameters) : GrpPerm ->
ToddCoxeterSchreier(G: parameters) : GrpPerm ->
SolubleSchreier(G: parameters) : GrpPerm ->
Verify(G: parameters ) : GrpPerm ->
Example GrpPerm_BSGS (H65E42)
Example GrpPerm_BSFS-2 (H65E43)
Defining Values for Attributes
AssertAttribute(G, "Order", n) : GrpPerm, MonStgElt, RngIntElt ->
AssertAttribute(G, "Order", Q) : GrpPerm, MonStgElt, [<RngIntElt, RngIntElt>] ->
[Future release] AssertAttribute(G, "BSGS", S) : GrpPerm, MonStgElt, GrpPermBSGS ->
Example GrpPerm_RandomSchreier (H65E44)
Accessing the Base and Strong Generating Set
Base(G) : GrpPerm -> [Elt]
BasePoint(G, i) : GrpPerm, RngIntElt -> Elt
BasicOrbit(G, i) : GrpPerm, RngIntElt -> SetIndx
BasicOrbits(G) : GrpPerm -> [SetIndx]
BasicOrbitLength(G, i) : GrpPerm, RngIntElt -> RngIntElt
BasicOrbitLengths(G) : GrpPerm -> [RngIntElt]
BasicStabilizer(G, i) : GrpPerm, RngIntElt -> GrpPerm
BasicStabilizerChain(G) : GrpPerm -> [GrpPerm]
IsMemberBasicOrbit(G, i, a) : GrpPerm, RngIntElt, Elt -> BoolElt
NumberOfStrongGenerators(G) : GrpPerm -> RngIntElt
NumberOfStrongGenerators(G, i) : GrpPerm, RngIntElt -> RngIntElt
SchreierVectors(G) : GrpPerm -> [ [RngIntElt] ]
SchreierVector(G, i) : GrpPerm, RngIntElt -> [RngIntElt]
StrongGenerators(G) : GrpPerm -> SetIndx(GrpPermElt)
StrongGenerators(G, i) : GrpPerm, RngIntElt -> SetIndx(GrpPermElt)
Working with a Base and Strong Generating Set
BaseImage(x) : GrpPermElt -> [Elt]
Permutation(G, Q) : GrpPerm, [Elt] -> GrpPermElt
SVPermutation(G, i, a) : GrpPerm, RngIntElt, Elt -> GrpPermElt
SVWord(G, i, a) : GrpPerm, RngIntElt, Elt -> GrpFPElt
Strip(H, x) : GrpPerm, GrpPermElt -> BoolElt, GrpPermElt, RngIntElt
WordStrip(H, x) : GrpPerm, GrpPermElt -> BoolElt, GrpFPElt, RngIntElt
BaseImageWordStrip(H, x) : GrpPerm, GrpPermElt -> BoolElt, GrpFPElt, RngIntElt
WordInStrongGenerators(H, x) : GrpPerm, GrpPermElt -> GrpFPElt
Modifying a Base and Strong Generating Set
ChangeBase(~G, Q) : GrpPerm, [Elt] ->
AddNormalizingGenerator(~H, x) : GrpPerm, GrpPermElt ->
ReduceGenerators(~G) : GrpPerm ->
Permutation Representations of Linear Groups
AffineGeneralLinearGroup(arguments)
AffineSpecialLinearGroup(arguments)
AffineGammaLinearGroup(arguments)
AffineSigmaLinearGroup(arguments)
AffineSymplecticGroup(arguments)
AffineSigmaSymplecticGroup(arguments)
ProjectiveGeneralLinearGroup(arguments)
ProjectiveSpecialLinearGroup(arguments)
ProjectiveGammaLinearGroup(arguments)
ProjectiveSigmaLinearGroup(arguments)
ProjectiveGeneralUnitaryGroup(arguments)
ProjectiveSpecialUnitaryGroup(arguments)
ProjectiveGammaUnitaryGroup(arguments)
ProjectiveSigmaUnitaryGroup(arguments)
ProjectiveSymplecticGroup(arguments)
ProjectiveSigmaSymplecticGroup(arguments)
PGO(arguments)
PGOPlus(arguments)
PGOMinus(arguments)
PSO(arguments)
PSOPlus(arguments)
PSOMinus(arguments)
ProjectiveOmega(arguments)
ProjectiveOmegaPlus(arguments)
ProjectiveOmegaMinus(arguments)
ProjectiveSuzukiGroup(arguments)
AffineGroup(M) : GrpMat[FldFin] -> GrpPerm, {@ ModTupFldElt @}
Permutation Group Databases
Ordering of Permutation Groups
CanonicalGenerators(G) : GrpPerm -> SeqEnum[GrpPermElt], GrpPermElt
CanonicalInvariant(G) : GrpPerm -> SeqEnum[SeqEnum], Seqenum
Example GrpPerm_CanonicalGenerators (H65E45)
Ordered Partition Stacks
Construction of Ordered Partition Stacks
OrderedPartitionStack(n) : RngIntElt -> StkPtnOrd
OrderedPartitionStackZero(n, h) : RngIntElt, RngIntElt -> StkPtnOrd
Properties of Ordered Partition Stacks
Degree(P) : StkPtnOrd -> RngIntElt
Height(P) : StkPtnOrd -> RngIntElt
NumberOfCells(P, h) : StkPtnOrd, RngIntElt -> RngIntElt
CellNumber(P, h, x) : StkPtnOrd, RngIntElt, RngIntElt -> RngIntElt
CellSize(P, h, i) : StkPtnOrd, RngIntElt, RngIntElt -> RngIntElt
Cell(P, h, i): StkPtnOrd, RngIntElt, RngIntElt -> SeqEnum
Random(P, i) : StkPtnOrd, RngIntElt -> RngIntElt
Representative(P, i) : StkPtnOrd, RngIntElt -> RngIntElt
ParentCell(P, i) : StkPtnOrd, RngIntElt -> RngIntElt
Operations on Ordered Partition Stacks
SplitCell(P, i, x) : StkPtnOrd, RngIntElt, RngIntElt -> BoolElt
SplitAllByValues(P, V) : StkPtnOrd, SeqEnum[RngIntElt] -> BoolElt, RngIntElt
SplitCellsByValues(P, C, V) : StkPtnOrd, SeqEnum[RngIntElt], SeqEnum[RngIntElt] -> BoolElt, RngIntElt
Pop(P) : StkPtnOrd ->
Advance(X, L, P, h) : StkPtnOrd, seqEnum[RngIntElt], StkPtnOrd, RngIntElt ->
Example GrpPerm_OrderedPartitionStack (H65E46)
Bibliography
[Next][Prev] [Right] [____] [Up] [Index] [Root]
|