# loops : a GAP 4 package - Index

A B C D E F G H I L M N O P Q R S T U

## A

A typical library 9.1
About Cayley tables 4.1
Acknowledgement 1.5
Additional varieties of loops 7.7
AllLoopsWithMltGroup 8.4.1
AllLoopTablesInGroup 8.4.1
AllProperLoopTablesInGroup 8.4.1
AllSubloops 6.3.3
AreEqualDiscriminators 6.13.2
AssociatedLeftBruckLoop 8.1.1
Associativity, commutativity and generalizations 7.1
Associator 5.4.1
Associators and commutators 5.4
AssociatorSubloop 6.7.5
Automorphic loops 9.10
AutomorphicLoop 9.10.1
AutomorphismGroup 6.12.3

## B

Basic arithmetic operations 5.2
Basic attributes 5.1
Basic methods and attributes 5.0

## C

Calculating with quasigroups 3.3
Canonical and normalized Cayley tables 4.3
CanonicalCayleyTable 4.3.1
CayleyTable 5.1.2
CayleyTableByPerms 4.6.1
CC-loops 9.6
CCLoop 9.6.1
Center 6.7.4
Code loops 9.4
CodeLoop 9.4.1
Commutant 6.7.3
Commutator 5.4.1
Comparing quasigroups with common parent 6.2
Conjugacy closed loops and related properties 7.6
Conversions 4.9
Conversions between magmas, quasigroups, loops and groups 3.2
Core methods for Bol loops 8.1
Creating quasigroups and loops 4.0
Creating quasigroups and loops by extensions 4.7
Creating quasigroups and loops by sections 4.6
Creating quasigroups and loops from a file 4.5
Creating quasigroups and loops manually 4.4

## D

DerivedLength 6.11.1
DerivedSubloop 6.11.1
DirectProduct 4.10.1
Discriminator 6.13.1
DisplayLibraryInfo 9.1.3
Documentation 1.2

## E

Elements 5.1.1
Exponent 5.1.6
Extensions 2.4

## F

Factor loops 6.9
FactorLoop 6.9.1
Feedback 1.4
Files A.0
Filters built into the package B.0
FrattinifactorSize 6.11.1
FrattiniSubloop 6.11.1

## G

Generators 5.5
GeneratorsOfLoop 5.5.1
GeneratorsOfQuasigroup 5.5.1
GeneratorsSmallest 5.5.2

## H

HasAntiautomorphicInverseProperty 7.2.3
HasAutomorphicInverseProperty 7.2.3
HasInverseProperty 7.2.1
HasLeftInverseProperty 7.2.1
HasRightInverseProperty 7.2.1
HasTwosidedInverses 7.2.1
HasWeakInverseProperty 7.2.2
Homomorphisms and homotopisms 2.3
How are isomorphisms computed 6.13
How the package works 3.0

## I

Inner mapping groups 6.6
InnerMappingGroup 6.6.1
Installation 1.1
Interesting loops 9.11
InterestingLoop 9.11.1
IntoGroup 4.9.4
IntoLoop 4.9.3
IntoQuasigroup 4.9.1
Introduction 1.0
Inverse 5.3.1
Inverse properties 7.2
IsALoop 7.7.5
IsAlternative 7.4.1
IsAssociative 7.1.1
IsAutomorphicLoop 7.7.4
IsCCLoop 7.6.1
IsCLoop 7.4.1
IsCodeLoop 7.7.1
IsCommutative 7.1.1
IsDiassociative 7.1.2
IsDistributive 7.3.3
IsEntropic 7.3.3
IsExtraLoop 7.4.1
IsFlexible 7.4.1
IsIdempotent 7.3.2
IsLCCLoop 7.6.1
IsLCLoop 7.4.1
IsLDistributive 7.3.4
IsLeftALoop 7.7.5
IsLeftAlternative 7.4.1
IsLeftAutomorphicLoop 7.7.4
IsLeftBolLoop 7.4.1
IsLeftBruckLoop 7.7.3
IsLeftDistributive 7.3.3
IsLeftKLoop 7.7.3
IsLeftNuclearSquareLoop 7.4.1
IsLeftPowerAlternative 7.5.1
IsLoopCayleyTable 4.2.2
IsLoopTable 4.2.2
IsMedial 7.3.3
IsMiddleALoop 7.7.5
IsMiddleAutomorphicLoop 7.7.4
IsMiddleNuclearSquareLoop 7.4.1
IsMoufangLoop 7.4.1
IsNilpotent 6.10.1
IsNormal 6.8.1
IsNuclearSquareLoop 7.4.1
IsomorphicCopyByNormalSubloop 6.12.5
IsomorphicCopyByPerm 6.12.4
IsomorphismLoops 6.12.1
Isomorphisms and automorphisms 6.12
IsOsbornLoop 7.6.2
IsotopismLoops 6.14.1
Isotopisms 6.14
IsPowerAlternative 7.5.1
IsPowerAssociative 7.1.2
IsQuasigroupCayleyTable 4.2.1
IsQuasigroupTable 4.2.1
IsRCCLoop 7.6.1
IsRCLoop 7.4.1
IsRDistributive 7.3.4
IsRightALoop 7.7.5
IsRightAlternative 7.4.1
IsRightAutomorphicLoop 7.7.4
IsRightBolLoop 7.4.1
IsRightBruckLoop 7.7.3
IsRightDistributive 7.3.3
IsRightKLoop 7.7.3
IsRightNuclearSquareLoop 7.4.1
IsRightPowerAlternative 7.5.1
IsSemisymmetric 7.3.1
IsSimple 6.8.3
IsSolvable 6.11.1
IsSteinerLoop 7.7.2
IsSteinerQuasigroup 7.3.2
IsStronglyNilpotent 6.10.2
IsSubloop 6.3.2
IsSubquasigroup 6.3.2
IsTotallySymmetric 7.3.1
IsUnipotent 7.3.2
ItpSmallLoop 9.12.1

## L

Left Bol loops 9.2
LeftBolLoop 9.2.1
LeftDivision 5.2.2
LeftDivisionCayleyTable 5.2.3
LeftInnerMapping 6.6.1
LeftInnerMappingGroup 6.6.1
LeftInverse 5.3.1
LeftMultiplicationGroup 6.5.1
LeftNucleus 6.7.1
LeftSection 6.4.2
LeftTranslation 6.4.1
Libraries of loops up to isotopism 9.12
Libraries of small loops 9.0
LibraryLoop 9.1.2
LoopByCayleyTable 4.4.2
LoopByCyclicModification 8.2.1
LoopByDihedralModification 8.2.2
LoopByExtension 4.7.2
LoopByLeftSection 4.6.2
LoopByRightSection 4.6.4
LoopFromFile 4.5.1
LoopMG2 8.2.3
Loops of Bol-Moufang type 7.4
LoopsUpToIsomorphism 6.12.2
LoopsUpToIsotopism 6.14.2
LowerCentralSeries 6.10.4

## M

Mathematical background 2.0
Methods based on permutation groups 6.0
MiddleInnerMapping 6.6.1
MiddleInnerMappingGroup 6.6.1
MiddleNucleus 6.7.1
Moufang loops 9.3
Moufang modifications 8.2
MoufangLoop 9.3.1
Multiplication groups 6.5
MultiplicationGroup 6.5.1
MultiplicativeNeutralElement 5.1.4
MyLibraryLoop 9.1.1

## N

Naming, viewing and printing quasigroups and their elements 3.4
NaturalHomomorphismByNormalSubloop 6.9.2
Nilpotency and central series 6.10
NilpotencyClassOfLoop 6.10.1
Nilpotent loops 9.9
NilpotentLoop 9.9.1
Normal subloops 6.8
NormalClosure 6.8.2
NormalizedQuasigroupTable 4.3.2
Nuc 6.7.1
NuclearExtension 4.7.1
Nuclei, commutant, center, and associator subloop 6.7
NucleusOfLoop 6.7.2
NucleusOfQuasigroup 6.7.2

## O

One 5.1.3
OneLoopTableInGroup 8.4.1
OneLoopWithMltGroup 8.4.1
OneProperLoopTableInGroup 8.4.1
Opposite 4.11.1
Opposite quasigroups and loops 4.11

## P

Paige loops 9.8
PaigeLoop 9.8.1
Parent 6.1.1
Parent of a quasigroup 6.1
PosInParent 6.1.3
Position 6.1.2
Power alternative loops 7.5
Powers and inverses 5.3
PrincipalLoopIsotope 4.9.2
Products of loops 4.10

## Q

QuasigroupByCayleyTable 4.4.1
QuasigroupByLeftSection 4.6.2
QuasigroupByRightSection 4.6.4
QuasigroupFromFile 4.5.1
Quasigroups and loops 2.1

## R

Random quasigroups and loops 4.8
RandomLoop 4.8.2
RandomNilpotentLoop 4.8.4
RandomQuasigroup 4.8.3
Realizing groups as multiplication groups of loops 8.4
RelativeLeftMultiplicationGroup 6.5.2
RelativeMultiplicationGroup 6.5.2
RelativeRightMultiplicationGroup 6.5.2
Representing quasigroups 3.1
RightCosets 6.3.4
RightDivision 5.2.2
RightDivisionCayleyTable 5.2.3
RightInnerMapping 6.6.1
RightInnerMappingGroup 6.6.1
RightInverse 5.3.1
RightMultiplicationGroup 6.5.1
RightNucleus 6.7.1
RightSection 6.4.2
RightTranslation 6.4.1
RightTransversal 6.3.5

## S

SetLoopElmName 3.4.1
SetQuasigroupElmName 3.4.1
Size 5.1.5
Small loops 9.7
SmallLoop 9.7.1
Solvability 6.11
Some properties of quasigroups 7.3
Specific methods 8.0
Steiner loops 9.5
SteinerLoop 9.5.1
Subloop 6.3.1
Subquasigroup 6.3.1
Subquasigroups and subloops 6.3

## T

Test files 1.3
Testing Cayley tables 4.2
Testing properties of quasigroups and loops 7.0
Translations 2.2
Translations and sections 6.4
Triality for Moufang loops 8.3
TrialityPcGroup 8.3.2
TrialityPermGroup 8.3.1

## U

UpperCentralSeries 6.10.3

[Up]