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 (H58E1)
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 (H58E2)
Construction of a General Permutation Group
PermutationGroup< X | L > : Set, List -> GrpPerm
PermutationGroup< n | L > : RngIntElt, List -> GrpPerm
Example GrpPerm_Hessian (H58E3)
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 (H58E4)
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 (H58E5)
Homomorphisms
hom<G | 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 (H58E6)
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 (H58E7)
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 (H58E8)
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 (H58E9)
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 (H58E10)
Example GrpPerm_SetOperations-2 (H58E11)
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 (H58E12)
Example GrpPerm_Classes-2 (H58E13)
Construction of a Subgroup
sub<G | L> : GrpPerm, List -> GrpPerm
ncl<G | L> : GrpPerm, List -> GrpPerm
Example GrpPerm_Constructors (H58E14)
Example GrpPerm_Constructors-2 (H58E15)
Example GrpPerm_Constructors-3 (H58E16)
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 -> 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 (H58E17)
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 (H58E18)
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
Example GrpPerm_Subgroups (H58E19)
Example GrpPerm_Subgroups-2 (H58E20)
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> ]
Construction of Quotient Groups
quo<G | L> : GrpPerm, List -> GrpPerm, Map
G / N : GrpPerm, GrpPerm -> GrpPerm
Example GrpPerm_Quotient (H58E21)
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, RngIntElt -> GrpPC, Map, SeqEnum, MonStgElt
Example GrpPerm_SpecialQuotient (H58E22)
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 (H58E23)
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
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 (H58E24)
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 (H58E25)
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 (H58E26)
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 ]
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 (H58E27)
Example GrpPerm_BlocksActions-2 (H58E28)
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) : GrpPerm, GrpPermElt -> GrpPermElt
JellyfishPreimage(G, x) : 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
FittingSubgroup(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 (H58E29)
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 (H58E30)
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 (H58E31)
The Socle
Socle(G) : GrpPerm -> GrpPerm
SocleFactor(G) : GrpPerm -> GrpPerm
SocleFactors(G) : GrpPerm -> [ GrpPerm ]
SocleSeries(G) : GrpPerm -> [ GrpPerm ]
EARNS(G) : 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 (H58E32)
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 (H58E33)
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 (H58E34)
Abelian Normal Subgroups
AbelianNormalSubgroup(G) : GrpPerm -> GrpPerm
AbelianNormalQuotient(G, H) : 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
H * g : GrpPerm, GrpPermElt -> Elt
DoubleCoset(G, H, g, K) : GrpPerm, GrpPerm, GrpPermElt, GrpPerm -> GrpPermDcosElt
DoubleCosetRepresentatives(G, H, K) : GrpPerm, GrpPerm, 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 atbrace, Map
TransversalProcess(G, H) : GrpPerm, GrpPerm -> GrpPermTransProc
TransversalProcessRemaining(P) : GrpPermTransProc -> RngIntElt
TransversalProcessNext(P) : GrpPermTransProc -> GrpPermElt
ShortCosets(p, H, G) : GrpPermElt, GrpPerm, GrpPerm -> [GrpPermElt]
Generators and Relations
FPGroup(G) : GrpPerm :-> GrpFP, Hom(Grp)
FPQuotient(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 (H58E35)
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 (H58E36)
Example GrpPerm_Cohomology-2 (H58E37)
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 (H58E38)
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) : GrpPerm -> BoolElt
TwoTransitiveGroupIdentification(G) : GrpPerm -> Tup
RecogniseAlternatingOrSymmetric(G, n) : Grp, RngIntElt -> BoolElt, BoolElt, UserProgram, UserProgram
IsEven(G): GrpPerm -> BoolElt
Example GrpPerm_RecogniseAltsym1 (H58E39)
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 (H58E40)
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 ) : RngIntElt ->
Example GrpPerm_BSGS (H58E41)
Example GrpPerm_BSFS-2 (H58E42)
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 (H58E43)
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)
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, { at ModTupFldElt atbrace
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 (H58E44)
Bibliography
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Mon Dec 17 14:40:36 EST 2012