# 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 W

## A

A typical library 8.1
About Cayley tables 4.1
Additional varieties of loops 6.6
antiautomorphic inverse property 6.2
AreEqualDiscriminators 5.13.2
AsGroup 4.6.4
AsLoop 4.6.2
AsQuasigroup 4.6.1
Associativity, commutativity and generalizations 6.1
Associator 4.12.1
associator 4.12
associator subloop 5.7
Associators and commutators 4.12
AssociatorSubloop 5.7.6
automorphic inverse property 6.2
AutomorphismGroup 5.12.2

## B

Basic arithmetic operations 4.10
Basic attributes 4.9
Basic methods and attributes 4.0

## C

C-loops 6.4
Calculating with quasigroups 3.3
Canonical and normalized Cayley tables 4.3
CanonicalCayleyTable 4.3.1
Cayley table 4.1
CayleyTable 4.9.2
CC-loops 8.5
CCLoop 8.5.1
Center 5.7.5
Commutant 5.7.4
commutant 5.7
Commutator 4.12.1
commutator 4.12
Comparing quasigroups with common parent 5.2
Conjugacy closed loops and related properties 6.5
Conversions 4.6
Conversions between magmas, quasigroups, loops and groups 3.2
Creating quasigroups and loops from a file 4.5
Creating quasigroups and loops manually 4.4
cyclic modification 7.1

## D

DerivedLength 5.11.1
DerivedSubloop 5.11.1
dihedral modification 7.1
DirectProduct 4.7.1
Discriminator 5.13.1
DisplayLibraryInfo 8.1.3
distributive quasigroup 6.3
Documentation 1.2

## E

Elements 4.9.1
Exponent 4.9.6
exponent 4.9
extra loops 6.4

## F

Factor loops 5.9
FactorLoop 5.9.1
Feedback 1.4
Files A.0
Filters built into the package B.0
flexible loops 6.4
FrattinifactorSize 5.11.1
FrattiniSubloop 5.11.1

## G

Generators 4.13
GeneratorsOfLoop 4.13.1
GeneratorsOfQuasigroup 4.13.1
GeneratorsSmallest 4.13.2
group 2.1
group with triality 7.2
groupoid 2.1

## H

HasAntiautomorphicInverseProperty 6.2.3
HasAutomorphicInverseProperty 6.2.3
HasInverseProperty 6.2.1
HasLeftInverseProperty 6.2.1
HasRightInverseProperty 6.2.1
HasTwosidedInverses 6.2.1
HasWeakInverseProperty 6.2.2
homomorphism 2.3
Homomorphisms and homotopisms 2.3
homotopism 2.3
How are isomorphisms computed 5.13
How the package works 3.0

## I

identity element 2.1
identity of Bol-Moufang type 6.4
inner mapping group 5.6
Inner mapping groups 5.6
InnerMappingGroup 5.6.1
Installation 1.1
Interesting loops 8.7
InterestingLoop 8.7.1
Introduction 1.0
Inverse 4.11.1
inverse 4.11
Inverse properties 6.2
inverse property 6.2
IsAlternative 6.4.1
IsAssociative 6.1.1
IsCCLoop 6.5.1
IsCLoop 6.4.1
IsCommutative 6.1.1
IsDiassociative 6.1.2
IsDistributive 6.3.3
IsEntropic 6.3.3
IsExtraLoop 6.4.1
IsFlexible 6.4.1
IsIdempotent 6.3.2
IsLCCLoop 6.5.1
IsLCLoop 6.4.1
IsLDistributive 6.3.4
IsLeftAlternative 6.4.1
IsLeftBolLoop 6.4.1
IsLeftBruckLoop 6.6.1
IsLeftDistributive 6.3.3
IsLeftKLoop 6.6.1
IsLeftNuclearSquareLoop 6.4.1
IsLoopCayleyTable 4.2.2
IsLoopTable 4.2.2
IsMedial 6.3.3
IsMiddleNuclearSquareLoop 6.4.1
IsMoufangLoop 6.4.1
IsNilpotent 5.10.1
IsNormal 5.8.1
IsNuclearSquareLoop 6.4.1
isomorphism 2.3
IsomorphismLoops 5.12.1
Isomorphisms and automorphisms 5.12
IsomorphismTypeOfMoufangLoop 8.3.2
IsOsbornLoop 6.5.2
isotopism 2.3
IsPowerAssociative 6.1.2
IsQuasigroupCayleyTable 4.2.1
IsQuasigroupTable 4.2.1
IsRCCLoop 6.5.1
IsRCLoop 6.4.1
IsRDistributive 6.3.4
IsRightAlternative 6.4.1
IsRightBolLoop 6.4.1
IsRightBruckLoop 6.6.1
IsRightDistributive 6.3.3
IsRightKLoop 6.6.1
IsRightNuclearSquareLoop 6.4.1
IsSemisymmetric 6.3.1
IsSimple 5.8.3
IsSolvable 5.11.1
IsSteinerLoop 6.6.2
IsSteinerQuasigroup 6.3.2
IsStronglyNilpotent 5.10.2
IsSubloop 5.3.2
IsSubquasigroup 5.3.2
IsTotallySymmetric 6.3.1
IsUnipotent 6.3.2
iterated centers 5.10

## L

Latin square 4.1
LC-loops 6.4
left alternative loops 6.4
Left Bol loops 8.2
left Bol loops 6.4
left division 4.10
left inner mapping group 5.6
left inverse 4.11
left inverse property 6.2
left multiplication group 2.2
left nuclear square loops 6.4
left nucleus 5.7
left section 2.2
left translation 2.2
LeftBolLoop 8.2.1
LeftDivision 4.10.2
LeftInnerMappingGroup 5.6.1
LeftInverse 4.11.1
LeftMultiplicationGroup 5.5.1
LeftNucleus 5.7.1
LeftSection 5.4.2
LeftTranslation 5.4.1
Libraries of small loops 8.0
LibraryLoop 8.1.2
list of files A.0
loop 2.1
loop table 4.1
LoopByCayleyTable 4.4.2
LoopByCyclicModification 7.1.1
LoopByDihedralModification 7.1.2
LoopFromFile 4.5.1
LoopMG2 7.1.3
Loops of Bol-Moufang type and related properties 6.4
lower central series 5.10
LowerCentralSeries 5.10.4

## M

magma 2.1
Mathematical background 2.0
middle nuclear square loops 6.4
middle nucleus 5.7
MiddleNucleus 5.7.1
monoid 2.1
Moufang center 5.7
Moufang loops 6.4
Moufang modifications 7.1
MoufangLoop 8.3.1
multiplication group 2.2
Multiplication groups 5.5
multiplication table 4.1
MultiplicationGroup 5.5.1
MultiplicativeNeutralElement 4.9.4
MyLibraryLoop 8.1.1

## N

Naming, viewing and printing quasigroups and their elements 3.4
NaturalHomomorphismByNormalSubloop 5.9.2
neutral element 2.1
Nilpotency and central series 5.10
NilpotencyClassOfLoop 5.10.1
normal closure 5.8
Normal subloops 5.8
NormalClosure 5.8.2
NormalizedQuasigroupTable 4.3.2
Nuc 5.7.1
Nuclei, commutant, center, and associator subloop 5.7
nucleus 5.7
NucleusOfLoop 5.7.2
NucleusOfQuasigroup 5.7.3

## O

octonion loop 8.3
octonions 8.3
One 4.9.3
Opposite 4.8.1
opposite quasigroup 4.8
Opposite quasigroups and loops 4.8

## P

Paige loops 8.6
PaigeLoop 8.6.1
Parent 5.1.1
Parent of a quasigroup 5.1
PosInParent 5.1.3
Position 5.1.2
power-associative 4.11
power-associative loop 4.9
Powers and inverses 4.11
principal isotopism 2.3
principal loop isotope 2.3
PrincipalLoopIsotope 4.6.3
Products of loops 4.7

## Q

quasigroup 2.1
quasigroup table 4.1
QuasigroupByCayleyTable 4.4.1
QuasigroupFromFile 4.5.1
Quasigroups and loops 2.1

## R

RC-loops 6.4
relative left multiplication group 5.5
relative multiplication group 5.5
relative right multiplication group 5.5
RelativeLeftMultiplicationGroup 5.5.2
RelativeMultiplicationGroup 5.5.2
RelativeRightMultiplicationGroup 5.5.2
Representing quasigroups 3.1
right alternative loops 6.4
right Bol loops 6.4
right division 4.10
right inner mapping group 5.6
right inverse 4.11
right inverse property 6.2
right multiplication group 2.2
right nuclear square loops 6.4
right nucleus 5.7
right section 2.2
right translation 2.2
RightDivision 4.10.2
RightInnerMappingGroup 5.6.1
RightInverse 4.11.1
RightMultiplicationGroup 5.5.1
RightNucleus 5.7.1
RightSection 5.4.2
RightTranslation 5.4.1

## S

semigroup 2.1
SetLoopElmName 3.4.1
SetQuasigroupElmName 3.4.1
Size 4.9.5
Small Moufang loops 8.3
Solvability 5.11
Some methods based on permutation groups 5.0
Some properties of quasigroups 6.3
Specific methods 7.0
Steiner loop 6.6
Steiner loops 8.4
SteinerLoop 8.4.1
Subloop 5.3.1
Subquasigroup 5.3.1
Subquasigroups and subloops 5.3

## T

Test files 1.3
Testing Cayley tables 4.2
Testing properties of quasigroups and loops 6.0
Translations 2.2
Translations and sections 5.4
Triality for Moufang loops 7.2
TrialityPcGroup 7.2.2
TrialityPermGroup 7.2.1
two-sided inverse 2.1
two-sided inverses 6.2

## U

upper central series 5.10
UpperCentralSeries 5.10.3

## W

weak inverse property 6.2

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