FINITE SOLUBLE GROUPS
Acknowledgements
Introduction
Power-Conjugate Presentations
Creation of a Group
Construction Functions
Definition by Presentation
Possibly Inconsistent Presentations
Basic Group Properties
Infrastructure
Numerical Invariants
Predicates
Homomorphisms
New Groups from Existing
Elements
Definition of Elements
Arithmetic Operations on Elements
Properties of Elements
Predicates for Elements
Set Operations
Conjugacy
Subgroups
Definition of Subgroups by Generators
Membership and Coercion
Inclusion and Equality
Standard Subgroup Constructions
Properties of Subgroups
Predicates for Subgroups
Hall π-Subgroups and Sylow Systems
Conjugacy Classes of Subgroups
Quotient Groups
Construction of Quotient Groups
Abelian and p-Quotients
Normal Subgroups and Subgroup Series
Characteristic Subgroups
Subgroup Series
Series for p-groups
Normal Subgroups and Complements
Cosets
Coset Tables and Transversals
Action on a Coset Space
Automorphism Group
General Soluble Group
Lifting Algorithm
Lifting from the Automorphism Group of a Sylow p-subgroup
p-group
Isomorphism and Standard Presentations
Generating p-groups
Representation Theory
Central Extensions
Transfer Between Group Categories
Transfer to GrpPC
Transfer from GrpPC
More About Presentations
Conditioned Presentations
Structure Operations
Element Operations
Special Presentations
CompactPresentation
Optimizing Magma Code
PowerGroup
p-Groups of Tame Genus
Verbose Printing
Constructors
Direct Indecomposability
Genus
Isomorphism
Automorphism Groups
Canonical Labelling
Bibliography
Introduction
Power-Conjugate Presentations
Creation of a Group
Construction Functions
CyclicGroup(GrpPC, n) : Cat, RngIntElt -> GrpPC
AbelianGroup(GrpPC, Q) : Cat, [RngIntElt] -> GrpPC
DihedralGroup(GrpPC, n) : Cat, RngIntElt -> GrpPC
ExtraSpecialGroup(GrpPC, p, n : parameters) : Cat, RngIntElt, RngIntElt -> GrpPC
Example GrpPC_Standard (H70E1)
Definition by Presentation
PolycyclicGroup< x1, ..., xn | R : parameters > : List(Identifiers), List(GrpFPRel) -> GrpPC, Map
quo< GrpPC : F | R : parameters > : GrpFP, List(GrpFPRel) -> GrpPC, Map
Example GrpPC_PolycyclicGroup (H70E2)
Possibly Inconsistent Presentations
IsConsistent(G) : GrpPC -> BoolElt
Example GrpPC_IsConsistent (H70E3)
Basic Group Properties
Infrastructure
G . i : GrpPC, RngIntElt -> GrpPCElt
Generators(G) : GrpPC -> SetEnum
NumberOfGenerators(G) : GrpPC -> RngIntElt
PCGenerators(G) : GrpPC -> SetIndx
NumberOfPCGenerators(G) : GrpPC -> RngIntElt
PCPrimes(G) : GrpPC -> [RngIntElt]
Numerical Invariants
Order(G) : GrpPC -> RngIntElt
FactoredOrder(G) : GrpPC -> [<RngIntElt, RngIntElt>]
Exponent(G) : GrpPC -> RngIntElt
Predicates
IsAbelian(G) : GrpPC -> BoolElt
IsCyclic(G) : GrpPC -> BoolElt
IsElementaryAbelian(G) : GrpPC -> BoolElt
IsNilpotent(G) : GrpPC -> BoolElt
IsPerfect(G) : GrpPC -> BoolElt
IsSimple(G) : GrpPC -> BoolElt
IsSoluble(G) : GrpPC -> BoolElt
IsTrivial(G) : GrpPC -> BoolElt
IsSpecial(G) : GrpPC -> BoolElt
IsExtraSpecial(G) : GrpPC -> BoolElt
Example GrpPC_group-props (H70E4)
Homomorphisms
hom< G -> H | L > : GrpPC, GrpPC, List -> Map
IsHomomorphism(G, H, L) : GrpPC, GrpPC, SeqEnum -> BoolElt, Map
IdentityHomomorphism(G) : GrpPC -> Map
Kernel(f) : Map -> GrpPC
Homomorphisms(G, H) : GrpPC, GrpPC -> SeqEnum
Example GrpPC_pc_hom (H70E5)
New Groups from Existing
DirectProduct(G, H) : GrpPC, GrpPC -> GrpPC, [Map], [Map]
DirectProduct(Q) : [GrpPC] -> GrpPC, [ Map ], [ Map ]
Extension(G, H, f) : GrpPC, GrpPC, [Map] -> GrpPC
Extension(M, H) : ModGrp, GrpPC -> GrpPC
Extension(G, H, f, t) : GrpPC, GrpPC, [Map], [GrpPCElt] -> GrpPC
Extension(M, H, t) : ModGrp, GrpPC, [ModGrpElt] -> GrpPC
IsExtension(G, H, f) : GrpPC, GrpPC, [Map] -> BoolElt, GrpPC
WreathProduct(G, H) : GrpPC, GrpPC -> GrpPC
WreathProduct(G, H, f) : GrpPC, GrpPC, Map -> GrpPC
Example GrpPC_extension (H70E6)
Example GrpPC_cossey_hawkes (H70E7)
Elements
Definition of Elements
G ! Q : GrpPC, [RngIntElt] -> GrpPCElt
ElementToSequence(x) : GrpPCElt -> [RngIntElt]
Identity(G) : GrpPC -> GrpPCElt
Example GrpPC_elt-definition (H70E8)
Arithmetic Operations on Elements
g * h : GrpPCElt, GrpPCElt -> GrpPCElt
g *:= h : GrpPCElt, GrpPCElt -> GrpPCElt
g ^ n: GrpPCElt, RngIntElt -> GrpPCElt
g ^:= n: GrpPCElt, RngIntElt -> GrpPCElt
g / h : GrpPCElt, GrpPCElt -> GrpPCElt
g /:= h : GrpPCElt, GrpPCElt -> GrpPCElt
g ^ h : GrpPCElt, GrpPCElt -> GrpPCElt
g ^:= h : GrpPCElt, GrpPCElt -> GrpPCElt
(g1, ..., gn) : List(GrpPCElt) -> GrpPCElt
Properties of Elements
Order(x) : GrpPCElt -> RngIntElt
Parent(x) : GrpPCElt -> GrpPC
Predicates for Elements
g eq h : GrpPCElt, GrpPCElt -> BoolElt
g ne h : GrpPCElt, GrpPCElt -> BoolElt
IsIdentity(g) : GrpPCElt -> BoolElt
IsConjugate(G, g, h) : GrpPC, GrpPCElt, GrpPCElt -> BoolElt, GrpPCElt
Example GrpPC_elt_predicates (H70E9)
Set Operations
NumberingMap(G) : GrpPC -> Map
Random(G) : GrpPC -> GrpPCElt
RandomProcess(G) : GrpPC -> Process
Random(P) : Process -> GrpPCElt
Representative(G) : GrpPC -> GrpPCElt
Example GrpPC_set_ops (H70E10)
Example GrpPC_Set (H70E11)
Conjugacy
Class(H, g) : GrpPC, GrpPCElt -> { GrpPCElt }
ConjugacyClasses(G) : GrpPC -> [ <RngIntElt, RngIntElt, GrpPCElt> ]
ClassMap(G) : GrpPC -> Map
ClassRepresentative(G, x) : GrpPC, GrpPCElt -> GrpPCElt
ClassCentraliser(G, i) : GrpPC, RngIntElt -> GrpPCElt
IsConjugate(G, g, h) : GrpPC, GrpPCElt, GrpPCElt -> BoolElt, GrpPCElt
NumberOfClasses(G) : GrpPC -> RngIntElt
PowerMap(G) : GrpPC -> Map
Example GrpPC_class_map (H70E12)
Subgroups
Definition of Subgroups by Generators
sub<G | L> : GrpPC, List -> GrpPC, Map
ncl<G | L> : GrpPC, List -> GrpPC, Map
Example GrpPC_sub_creation (H70E13)
Membership and Coercion
g in G : GrpPCElt, GrpPC -> BoolElt
g notin G : GrpPCElt, GrpPC -> BoolElt
G ! g : GrpPC, GrpPCElt -> GrpPCElt
H ! g : GrpPC, GrpPCElt -> GrpPCElt
K ! g : GrpPC, GrpPCElt -> GrpPCElt
Example GrpPC_coercion (H70E14)
Inclusion and Equality
S subset G : { GrpPCElt } , GrpPC -> BoolElt
S notsubset G : { GrpPCElt } , GrpPC -> BoolElt
H subset G : GrpPC, GrpPC -> BoolElt
H notsubset G : GrpPC, GrpPC -> BoolElt
G eq H : GrpPC, GrpPC -> BoolElt
G ne H : GrpPC, GrpPC -> BoolElt
InclusionMap(G, H) : GrpPC, GrpPC -> Map
Standard Subgroup Constructions
H ^ g : GrpPC, GrpPCElt -> GrpPC
H meet K : GrpPC, GrpPC -> GrpPC
H meet:= K : GrpPC, GrpPC -> GrpPC
CommutatorSubgroup(G, H, K) : GrpPC, GrpPC, GrpPC -> GrpPC
Centralizer(G, g) : GrpPC, GrpPCElt -> GrpPC
Centralizer(G, H) : GrpPC, GrpPC -> GrpPC
Core(G, H) : GrpPC, GrpPC -> GrpPC
H ^ G : GrpPC, GrpPC -> GrpPC
Normalizer(G, H) : GrpPC, GrpPC -> GrpPC
Example GrpPC_subgroup-constructions (H70E15)
Properties of Subgroups
Index(G, H) : GrpPC, GrpPC -> RngIntElt
FactoredIndex(G, H) : GrpPC, GrpPC -> [<RngIntElt, RngIntElt>]
Predicates for Subgroups
IsCentral(G, H) : GrpPC, GrpPC -> BoolElt
IsConjugate(G, H, K) : GrpPC, GrpPC, GrpPC -> BoolElt, GrpPCElt
IsMaximal(G, H) : GrpPC, GrpPC -> BoolElt
IsNormal(G, H) : GrpPC, GrpPC -> BoolElt
IsSelfNormalizing(G, H) : GrpPC, GrpPC -> BoolElt
IsSubnormal(G, H) : GrpPC, GrpPC -> BoolElt
Example GrpPC_sub-predicates (H70E16)
Hall π-Subgroups and Sylow Systems
ComplementBasis(G) : GrpPC -> [GrpPC]
HallSubgroup(G, S) : GrpPC, { RngIntElt } -> GrpPC
pCore(G, S) : GrpPC, { RngIntElt } -> GrpPC
SylowBasis(G) : GrpPC -> [GrpPC]
SylowSubgroup(G, p) : GrpPC, RngIntElt -> GrpPC
SystemNormalizer(G) : GrpPC -> GrpPC
Example GrpPC_Hall (H70E17)
Conjugacy Classes of Subgroups
SubgroupClasses(G) : GrpPC -> SeqEnum
AbelianSubgroups(G) : GrpPC -> SeqEnum
LowIndexSubgroups(G, n) : GrpPC, RngIntElt -> []
MaximalSubgroups(G) : GrpPC -> [GrpPC]
SubgroupLattice(G) : GrpPC -> SubGrpLat
BurnsideMatrix(G) : GrpPC -> AlgMatElt
TableOfMarks(G) : GrpPC -> AlgMatElt
DisplayBurnsideMatrix(G) : GrpPC ->
Example GrpPC_SubgroupClasses (H70E18)
Quotient Groups
Construction of Quotient Groups
quo<G | L> : GrpPC, List -> GrpPC, Map
G / N : GrpPC, GrpPC -> GrpPC
Example GrpPC_pc_quotient (H70E19)
Abelian and p-Quotients
AbelianQuotient(G) : GrpPC -> GrpAb, Map
AbelianQuotientInvariants(G) : GrpPC -> SeqEnum
ElementaryAbelianQuotient(G, p) : GrpPC, RngIntElt -> GrpAb, Map
pQuotient(G, p, c : parameters ) : GrpPC, RngIntElt, RngIntElt -> GrpPC, Map
Normal Subgroups and Subgroup Series
Characteristic Subgroups
Centre(G) : GrpPC -> GrpPC
CommutatorSubgroup(G) : GrpPC -> GrpPC
FittingSubgroup(G) : GrpPC -> GrpPC
FrattiniSubgroup(G) : GrpPC -> GrpPC
FrattiniQuotientRank(G) : GrpPC -> GrpPC
Hypercentre(G) : GrpPC -> GrpPC
MinimalNormalSubgroups(G) : GrpPC -> [GrpPC]
pCore(G, S) : GrpPC, { RngIntElt } -> GrpPC
Socle(G) : GrpPC -> GrpPC
Subgroup Series
PrimaryAbelianBasis(G) : GrpPC -> [ GrpPCElt ], [ RngIntElt ]
PrimaryAbelianInvariants(G) : GrpPC -> [RngIntElt]
ChiefSeries(G) : GrpPC -> [GrpPC]
CompositionSeries(G) : GrpPC -> [GrpPC]
CompositionFactors(G) : GrpPC -> SeqEnum
CompositionSeries(G, i) : GrpPC, RngIntElt -> [GrpPC]
DerivedSeries(G) : GrpPC -> [GrpPC]
DerivedLength(G) : GrpPC -> RngIntElt
ElementaryAbelianSeries(G) : GrpPC -> [GrpPC]
ElementaryAbelianSeriesCanonical(G) : GrpPC -> [GrpPC]
LowerCentralSeries(G) : GrpPC -> [GrpPC]
NilpotencyClass(G) : GrpPC -> RngIntElt
pCentralSeries(G, p) : GrpPC, RngIntElt -> [GrpPC]
SubnormalSeries(G, H) : GrpPC, GrpPC -> [GrpPC]
UpperCentralSeries(G) : GrpPC -> [GrpPC]
Example GrpPC_EAS (H70E20)
Series for p-groups
Agemo(G, i) : GrpPC, RngIntElt -> GrpPC
Omega(G, i) : GrpPC, RngIntElt -> GrpPC
JenningsSeries(G) : GrpPC -> [GrpPC]
pClass(G) : GrpPC -> RngIntElt
pRanks(G) : GrpPC-> [ RngIntElt ]
Normal Subgroups and Complements
NormalSubgroups(G) : GrpPC -> SeqEnum
NormalLattice(G) : GrpPC -> SubGrpLat
MinimalNormalSubgroup(G) : GrpPC -> GrpPC
MinimalNormalSubgroup(G, N) : GrpPC -> GrpPC
Complements(G, N) : GrpPC, GrpPC -> SeqEnum
NormalComplements(G, N) : GrpPC, GrpPC -> SeqEnum
NormalComplements(G, H, N) : GrpPC, GrpPC, GrpPC -> SeqEnum
Example GrpPC_NormalComplements (H70E21)
Cosets
Coset Tables and Transversals
Transversal(G, H) : GrpPC, GrpPC -> {@ GrpPCElt @}, Map
CosetTable(G, H) : GrpPC, GrpPC -> Map
Transversal(G, H, K) : GrpPC, GrpPC, GrpPC -> {@ GrpPCElt @}, Map
ShortCosets(p, H, G) : GrpPCElt, GrpPC, GrpPC -> [GrpPCElt]
Action on a Coset Space
CosetAction(G, H) : Grp, Grp -> Hom(Grp), GrpPerm, GrpPC
CosetImage(G, H) : Grp, Grp -> GrpPerm
CosetKernel(G, H) : Grp, Grp -> Grp
Automorphism Group
General Soluble Group
Lifting Algorithm
AutomorphismGroup(G): GrpPC -> GrpAuto
HasAttribute(A, "GenWeights") : GrpAuto, MonStgElt -> BoolElt, [ RngIntElt ]
HasAttribute(A, "WeightSubgroupOrders") : GrpAuto, MonStgElt -> BoolElt, [ RngIntElt ]
Example GrpPC_AutomorphismGroup (H70E22)
Lifting from the Automorphism Group of a Sylow p-subgroup
AutomorphismGroupSolubleGroup(G: parameters): GrpPC -> GrpAuto
IsIsomorphicSolubleGroup(G, H: parameters) : GrpPC, GrpPC -> BoolElt, Map
Example GrpPC_AutomorphismGroupSolubleGroup (H70E23)
p-group
AutomorphismGroup(G: parameters): GrpPC -> GrpAuto
Example GrpPC_pAutomorphismGroup (H70E24)
OrderAutomorphismGroupAbelianPGroup (A) : SeqEnum -> RngIntElt
Example GrpPC_subgroupsabelianpgroups (H70E25)
Isomorphism and Standard Presentations
StandardPresentation(G): GrpPC -> GrpPC, Map
IsIdenticalPresentation(G, H) : GrpPC, GrpPC -> BoolElt
IsIsomorphic(G, H) : GrpPC, GrpPC -> BoolElt, Map, GrpPC
Example GrpPC_StandardPresentation (H70E26)
Generating p-groups
GeneratepGroups (p, d, c : parameters) : RngIntElt, RngIntElt,RngIntElt -> [GrpPC], RngIntElt
Descendants(G : parameters) : GrpPC -> [GrpPC], RngIntElt
Example GrpPC_Generating_p_groups (H70E27)
Example GrpPC_GeneratepGroups (H70E28)
Example GrpPC_IsGood (H70E29)
ClassTwo(p, d : parameters) : RngIntElt, RngIntElt -> SeqEnum
Example GrpPC_ClassTwo (H70E30)
Representation Theory
CharacterDegrees(G) : GrpPC -> [ Tup ]
CharacterDegrees(G) : GrpFin -> [ Tup ]
CharacterDegreesPGroup(G) : GrpFin -> [ RngIntElt ]
CharacterTable(G: parameters) : GrpPC -> TabChtr
CharacterTableConlon(G) : GrpPC -> [ AlgChtrElt ]
GModule(G, M) : GrpPC, AlgMat -> ModAlg
GModule(G, A) : GrpPC, GrpPC -> ModAlg, Map
GModule(G, A, B) : GrpPC, GrpPC, GrpPC -> ModAlg, Map
AbsolutelyIrreducibleRepresentationsSchur(G, k: parameters) : GrpPC, Rng -> List[Map]
IrreducibleRepresentationsSchur(G, k: parameters) : GrpPC, Rng -> List[Map]
Example GrpPC_Reps (H70E31)
Central Extensions
ExtGenerators(G, U) : GrpPC, GrpPC -> [<AlgMatElt, RngIntElt>]
HomGenerators(G, U) : GrpPC, GrpPC -> [<AlgMatElt, RngIntElt>]
ElementSequence(G) : GrpPC -> SeqEnum
RepresentativeCocycles(G, U, Ext, Hom) : GrpPC, GrpPC, [AlgMatElt], [AlgMatElt]-> [AlgMatElt]
CentralExtension(G, U, A) : GrpPC, GrpPC, AlgMatElt -> GrpPC
CentralExtensions(G, U, Q) : GrpPC, GrpPC, [AlgMatElt] -> [GrpPC]
CentralExtensionProcess(G, U) : GrpPC, GrpPC -> Proc
NextExtension(~P) : Rec -> GrpPC
IsEmpty(P) : Rec -> BoolElt
Example GrpPC_CentralExtension (H70E32)
Transfer Between Group Categories
Transfer to GrpPC
PCGroup(G) : GrpPerm -> GrpPC, Map
pQuotient(F, p, c : parameters ) : GrpFP, RngIntElt, RngIntElt -> GrpPC, Map
SolubleQuotient(G) : Grp -> GrpPC, Map
Example GrpPC_pcgroup (H70E33)
Transfer from GrpPC
AbelianGroup(G) : GrpPC -> GrpAb, Map
FPGroup(G) : GrpPC -> GrpFP, Map
GPCGroup(G) : GrpPC -> GrpGPC, Map
Example GrpPC_pc-to-perm (H70E34)
More About Presentations
Conditioned Presentations
Structure Operations
ConditionedGroup(G) : GrpPC -> GrpPC
IsConditioned(G) : GrpPC -> BoolElt
Element Operations
LeadingTerm(x) : GrpPCElt -> GrpPCElt
LeadingGenerator(x) : GrpPCElt -> GrpPCElt
LeadingExponent(x) : GrpPCElt -> RngIntElt
Depth(x) : GrpPCElt -> RngIntElt
PCClass(x) : GrpPCElt -> RngIntElt
Special Presentations
SpecialPresentation(G) : GrpPC -> GrpPC
SpecialWeights(G) : GrpPC -> [ <RngIntElt, RngIntElt, RngIntElt> ]
NilpotentLength(G) : GrpPC -> RngIntElt
NilpotentBoundary(G,i) : GrpPC, RngIntElt -> RngIntElt
MinorLength(G,i) : GrpPC, RngIntElt -> RngIntElt
MinorBoundary(G,i,j) : GrpPC, RngIntElt, RngIntElt -> RngIntElt
LayerLength(G,i,j) : GrpPC, RngIntElt, RngIntElt -> RngIntElt
LayerBoundary(G,i,j,k) : GrpPC, RngIntElt, RngIntElt, RngIntElt -> RngIntElt
Example GrpPC_SpecialPresentation (H70E35)
CompactPresentation
CompactPresentation(G) : GrpPC -> [RngIntElt]
PCGroup(Q : parameters ) : [RngIntElt] -> GrpPC
Example GrpPC_CompactPresentation (H70E36)
Optimizing Magma Code
PowerGroup
Example GrpPC_PowerGroupTwo (H70E37)
p-Groups of Tame Genus
Verbose Printing
Example GrpPC_VerbosePrinting (H70E38)
Constructors
TGRandomGroup(q, n, g : parameters) : RngIntElt, RngIntElt, RngIntElt -> GrpPC
Example GrpPC_RandomGenusGroups (H70E39)
RandomGenus2Group(q, d : parameters) : RngIntElt, [RngIntElt] -> GrpPC
Example GrpPC_PrescribedBlocks (H70E40)
RandomGenus1Group(q, d, r : parameters) : RngIntElt, RngIntElt, RngIntElt -> GrpPC
Example GrpPC_Heisenbergs (H70E41)
Genus2Group(f) : RngUPolElt -> GrpPC
Example GrpPC_Pfaffians (H70E42)
Direct Indecomposability
IsIndecomposable(G) : GrpPC -> BoolElt
IsIndecomposable(t) : TenSpcElt -> BoolElt
Example GrpPC_DecomposableGroups (H70E43)
Example GrpPC_DirectNotCentral (H70E44)
Genus
Genus(G) : GrpPC -> RngIntElt
Genus(t) : TenSpcElt -> RngIntElt
Example GrpPC_Genus (H70E45)
IsTameGenusGroup(G) : Group -> BoolElt
IsTameGenusTensor(t) : TenSpcElt -> BoolElt
Example GrpPC_NonExample (H70E46)
Example GrpPC_AllTheSmallGroups (H70E47)
Isomorphism
TGIsIsomorphic(G, H : parameters) : GrpPC, GrpPC -> BoolElt
Example GrpPC_IsomorphismTesting (H70E48)
TGIsPseudoIsometric(s, t : parameters) : TenSpcElt, TenSpcElt -> BoolElt, Hmtp
Example GrpPC_PseudoIsometries (H70E49)
Automorphism Groups
TGAutomorphismGroup(G : parameters) : GrpPC -> GrpAuto
Example GrpPC_FlatIndecomposable (H70E50)
TGPseudoIsometryGroup(t : parameters) : TenSpcElt -> GrpMat
Example GrpPC_Extensions (H70E51)
Canonical Labelling
TGSignature(G) : GrpPC -> List
TGSignature(t) : TenSpcElt -> List
Example GrpPC_ManyBlocks (H70E52)
Example GrpPC_MoreSmallGroups (H70E53)
Bibliography
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