LOOPS
Version 0.998
November 4, 2004
Package for GAP 4
Gábor P. Nagy
Department of Mathematics
University of Szeged
e-mail: nagyg@math.u-szeged.hu
Petr Vojtechovský
Department of Mathematics
University of Denver
e-mail: petr@math.du.edu

# Contents

1 Introduction
1.1 Installation
1.1.1 Brief description of `LOOPS` files
1.2 Documentation
1.3 Test files
1.4 Mathematical background
1.4.1 Quasigroups and loops
1.4.2 Translations
1.4.3 Homomorphisms and homotopisms
1.5 How the package works
1.5.1 Representing quasigroups in `LOOPS`
1.5.2 Calculating with quasigroups in `LOOPS`
1.5.3 Magmas, quasigroups, loops and groups in `GAP`
1.6 Feedback
2 Core methods
2.1 Naming, viewing and printing objects in `LOOPS`
2.1.1 Named quasigroups and loops
2.1.2 View mode
2.1.3 Print mode
2.2 Creating quasigroups and loops
2.2.1 Testing multiplication tables
2.2.2 Creating quasigroups and loops manually
2.2.3 Creating quasigroups and loops from a file
2.2.4 Conversions
2.2.5 Products of loops
2.2.6 Opposite quasigroups and loops
2.3 GAP categories
2.4 Basic attributes
2.5 Basic arithmetic operations
2.5.1 Multiplication
2.5.2 Division
2.5.3 Powers and inverses
2.5.4 Associators and commutators
2.6 Generators
2.7 Permutations associated with loops
2.7.1 Sections
2.7.2 Multiplication groups
2.7.3 Inner mapping group
2.8 Subquasigroups and subloops
2.9 Nucleus, commutant, center
2.10 Testing properties
2.10.1 Associativity, commutativity and generalizations
2.10.2 Inverse properties
2.10.3 Properties of quasigroups
2.10.4 Loops of Bol-Moufang type and related properties
2.10.5 Conjugacy closed loops
2.10.6 Additional varieties of loops
2.11 Normality
2.12 Factor loop
2.13 Nilpotency
2.14 Solvability
2.15 Filters built into `LOOPS`
3 Specific methods
3.1 Isomorphisms and automorphisms
3.1.1 Discriminator
3.2 Moufang modifications
3.2.1 Cyclic modification
3.2.2 Dihedral modification
3.2.3 Loops M(G,2)
3.3 Triality for Moufang loops
4 Libraries of small loops
4.1 A typical library
4.2 Left Bol loops
4.3 Small Moufang loops
4.3.1 Search for additional Moufang loops
4.4 Steiner loops
4.5 Paige loops
4.6 Interesting loops
5 Plans for future versions
5.1 Alternative representations of quasigroups and loops in `GAP`
5.2 Better support for quasigroups
5.3 More homomorphisms and homotopisms
5.4 Expanded libraries
5.5 Bits and pieces
Index

# Chapter 1
Introduction

`LOOPS` is a package for `GAP`4 whose purpose is to:

(a) provide researchers in nonassociative algebra with a powerful computational tool concerning finite loops and quasigroups,
(b) extend `GAP` toward the realm of nonassociative structures.

## 1.1 Installation

We assume that you have `GAP` v. 4.4 or newer installed on your computer. Download the `LOOPS` package from the distribution website

``http``:``/``/``www``.``math``.``du``.``edu``/``loops``

Unpack the downloaded file into the `pkg` subfolder of your `GAP` folder. After this step, there should be a subfolder `loops` in your `pkg` folder. The package `LOOPS` can then be loaded to `GAP` anytime by calling `LoadPackage("loops")`.
If you wish to load `LOOPS` automatically while starting `GAP`, proceed as follows:

(i) open the file `PackageInfo.g` in the `loops` folder,
(ii) edit the value of `Autoload` from `Autoload:=false` to `Autoload:=true`. (The parameter `Autoload` is located near the end of the file `PackageInfo.g`).

### 1.1.1 Brief description of `LOOPS` files

Table 1.1 summarizes all relevant files forming the `LOOPS` package. Some technical files (e.g., pictures used in the documentation) are not mentioned. You probably don't need any of this information unless you want to modify `LOOPS`.
Table 1.1: Brief description of files forming `LOOPS`.

folder

file

description

``pkg``

``README``.``loops``

installation and usage instructions

``pkg``/``loops``

``init``.``g``

declaration of ``LOOPS`` methods

``

``PackageInfo``.``g``

loading info for ``GAP`` 4.4

``

``read``.``g``

implementation of ``LOOPS`` methods

``pkg``/``loops``/``data``

``interesting``.``tbl``

library of interesting loops

``

``leftbol``.``tbl``

library of left Bol loops

``

``moufang``.``tbl``

library of Moufang loops

``

``moufang``_``discriminators``.``tbl``

data for isomorphisms of Moufang loops

``

``paige``.``tbl``

library of Paige loops

``

``steiner``.``tbl``

library of Steiner loops

``pkg``/``loops``/``doc``

``loops``_``manual``.``dvi``

dvi version of the documentation

``

``loops``_``manual``.``html``

web version of the documentation

``

``loops``_``manual``.``pdf``

pdf version of the documentation

``

``loops``_``manual``.``ps``

postscript version of the documentation

``

``loops``_``manual``.``tex``

LATEXsource for the documentation

``

``manual``.``six``

contextual help for ``GAP``

``pkg``/``loops``/``etc``

``

utilities for archive and doc generation

``pkg``/``loops``/``gap``

``banner``.``g``

banner of ``LOOPS``

``

``examples``.``gd``

methods for loop libraries

``

``loop``_``iso``.``gd``

methods for isomorphisms of loops

``

``moufang``_``modifications``.``gd``

methods for Moufang modifications

``

``quasigrp``.``gd``

core methods of ``LOOPS``

``

``triality``.``gd``

methods for triality of Moufang loops

``

``examples``.``gi``

methods for loop libraries

``

``loop``_``iso``.``gi``

methods for isomorphisms of loops

``

``moufang``_``modifications``.``gi``

methods for Moufang modifications

``

``quasigrp``.``gi``

core methods of ``LOOPS``

``

``triality``.``gi``

methods for triality of Moufang loops

``pkg``/``loops``/``tst``

``auto``.``tst``

test automorphism groups

``

``lib``.``tst``

test all libraries except Moufang

``

``mouflib``.``tst``

test library of Moufang loops

``

``nilpot``.``tst``

test nilpotency

``

``quasigrp``.``tst``

test core functions

``

``testall``.``g``

batch file for all tests

## 1.2 Documentation

The documentation is available is several formats: LATEX, pdf, dvi, postscript, html, and as an online help in `GAP`. All these formats have been obtained directly from the master LATEX documentation file. Consequently, the different formats differ only in their appearance, not in contents.
The documentation can be found in the `doc` folder of the `LOOPS` package and also at the `LOOPS` distribution website.
The online `GAP` help is available upon installing `LOOPS`, and can be accessed in the usual way, i.e., upon typing `?command`, `GAP` displays the section of the `LOOPS` manual containing information about `command`.

## 1.3 Test files

Test files conforming to the `GAP` standards are provided for `LOOPS`. They can be found in the folder `loops/tst` and run in the usual way.
The file `testall.tst` runs all tests for `LOOPS` with the exception of the test `mouflib.tst`. The test `mouflib.tst` builds all Moufang loops contained in the libraries of `LOOPS`, and runs for about 20 minutes.

## 1.4 Mathematical background

We assume that you are familiar with the theory of quasigroups and loops, for instance with the textbook of Bruck Bruck, [1971] or Pflugfelder Pflugfelder, [1990]. Nevertheless, we did include definitions and results in this manual in order to unify the terminology and improve the intelligibility of the text.

### 1.4.1 Quasigroups and loops

A set with one binary operation (denoted · here) is called groupoid or magma , the latter name being used in `GAP`. Associative groupoids are known as semigroups .
An element 1 of a groupoid G is a neutral element or an identity element if 1·x = x·1 = x for every x in G. Semigroup with a neutral element is a monoid .
Let G be a groupoid with neutral element 1. Then an element y is called a two-sided inverse of x in G if x·y = y·x = 1. A monoid in which every element has a two-sided inverse is called a group .
Groups can be reached in another way from groupoids, namely through quasigroups and loops.
A quasigroup Q is a groupoid such that the equation x·y=z has a unique solution in Q whenever two of the three elements x, y, z of Q are specified. Note that multiplication tables of finite quasigroups are precisely Latin squares , i.e., a square arrays with symbols arranged so that each symbol occurs in each row and in each column exactly once. A loop L is a quasigroup with a neutral element.
Groups are clearly loops, and one can show easily that an associative quasigroup is a group. Hence the theory of quasigroups and loops is in a sense complementary to the theory of semigroups and monoids.

### 1.4.2 Translations

Given an element x of a quasigroup Q we can associative two permutations of Q with it: the left translation Lx:Q® Q defined by y® x·y, and the right translation Rx:Q® Q defined by y® y·x.
Although it is possible to compose two right (left) translations, the resulting permutation is not necessarily a right (left) translation. The set L(Q)={Lx;x Î Q} is called the left section of Q, and, similarly, R(Q)={Rx;x Î Q} is the right section of Q.
Let SQ be the symmetric group on Q. Then LMlt(Q), the subgroup of SQ generated by L(Q), is called the left multiplication group of Q. Similarly, RMlt(Q) = á R(Q)ñ is the right multiplication group of Q. The smallest group containing both LMlt(Q) and RMlt(Q) is called the multiplication group of Q and is denoted by Mlt(Q).

### 1.4.3 Homomorphisms and homotopisms

Let K, H be two quasigroups. Then a map f:K® H is a homomorphism if f(x)·f(y)=f(x·y) for every x, y Î K. If f is also a bijection, we speak of an isomorphism , and the two quasigroups are called isomorphic.
The ordered triple (a,b,g) of maps a, b, g:K® H is a homotopism if a(x)·b(y) = g(x·y) for every x, y Î K. If the three maps are bijections, (a,b,g) is a isotopism , and the two quasigroups are isotopic.
Isotopic groups are necessarily isomorphic, but this is certainly not true for nonassociative quasigroups or loops. In fact, every quasigroup is isotopic to a loop, as we shall see.
Let (K,·), (K,°) be two quasigroups defined on the same set K. Then an isotopism (a,b,idK) is called a principal isotopism . An important class of principal isotopisms is obtained as follows:
Let (K,·) be a quasigroup, and let f, g be elements of K. Define a new operation ° on K by

x°y = Rg-1(x)·Lf-1(y),

where Rg, Lf are translations. Then (K,°) is a quasigroup isotopic to (K,·), in fact a loop with neutral element f·g. We call (K,°) a principal loop isotope of (K,·).

## 1.5 How the package works

The package consists of three complementary components: the core algorithms for quasigroup theoretical notions (see Chapter 2), some specific algorithms, mostly for Moufang loops (see Chapter 3), and the library of small loops (see Chapter 4).
Although we do not explain the algorithms in detail here, we describe the overarching ideas so that the user should be able to anticipate the capabilities and behavior of the computation.

### 1.5.1 Representing quasigroups in `LOOPS`

Since the permutation representation in the usual sense is impossible for nonassociative structures, and since the theory of nonassociative presentations is not well understood, we had to resort to multiplication tables to represent quasigroups in `GAP`. In order to save storage space, we sometimes use one multiplication table to represent several quasigroups (for instance when a quasigroup is a subquasigroup of another quasigroup).
Consequently, the package is intended primarily for quasigroups and loops of small order (up to 1000, say).

### 1.5.2 Calculating with quasigroups in `LOOPS`

Although the quasigroups are ultimately represented by multiplication tables, the algorithms are efficient because nearly all calculations are delegated to groups. The connection between quasigroup and groups is facilitated via the above-mentioned translations, and we illustrate it with a few examples:

1) This example shows how properties of quasigroups can be translated into properties of translations in a straightforward way.
Let Q be a quasigroup. We ask if Q is associative. We can either test if (xy)z=x(yz) for every x, y, z Î Q, or we can ask if Lxy=LxLy for every x, y Î Q. Note that since Lxy, Lx, Ly are elements of a permutation group, we do not have to refer directly to the multiplication table once the left translations of Q are known.
2) This example shows how properties of loops can be translated into properties of translations in a way that requires some theory.
A left Bol loop is a loop satisfying x(y(xz)) = (x(yx))z. We claim (without proof) that a loop L is left Bol if and only if LxLyLx is a left translation for every x, y Î L.
3) This example shows that many properties of loops become purely group-theoretical once they are expresses in terms of translations.
A loop is simple if it has no nontrivial congruences. Then it is easy to see that a loop is simple if and only if its multiplication group Mlt(L) is a primitive permutation group.

The main idea of the package is therefore: (i) calculate the translations and the associated permutation groups when they are needed, (ii) store them as attributes, (iii) use them in algorithms as often as possible.

### 1.5.3 Magmas, quasigroups, loops and groups in `GAP`

Whether an object is considered a quasigroup or a loop is a matter of declaration in `LOOPS`. A declared loop is considered to be a quasigroup, however, a declared quasigroup is not considered to be a loop, even if it accidentally possesses a neutral element. It is possible to convert a quasigroup Q (with or without a neutral element) to a loop using
· `AsLoop``( Q ) F`
The category of quasigroups (cf. `IsQuasigroup`) is declared in `LOOPS` so that it is contained in the category of magmas (cf. `IsMagma`). All standard `GAP` command for magmas are therefore available for quasigroups and loops, too.
Although groups are quasigroups mathematically, they are not treated as quasigroups in `LOOPS`. If you wish to apply methods of `LOOPS` to groups, apply one of the conversions
· `AsQuasigroup``( G ) F`
· `AsLoop``( G ) F`
to the group G. These conversions fail when G is infinite and will exhaust all available memory when G is huge. For more information on conversions, see subsection 2.2.4.

## 1.6 Feedback

We welcome all comments and suggestions on `LOOPS`, especially those concerning the future development of the package. You can contact us by e-mail.

# Chapter 2
Core methods

## 2.1 Naming, viewing and printing objects in `LOOPS`

`GAP` displays information about objects in two modes: `View` (default, short) and `Print` (longer). Moreover, when the name of an object is set, it is always shown, no matter which display mode is used.

### 2.1.1 Named quasigroups and loops

Only loops contained in the libraries of `LOOPS` are named. For instance, the loop obtained via `MoufangLoop( 32, 4 )`-the 4th Moufang loop of order 32-is named `<Moufang loop 32/4>`.

### 2.1.2 View mode

When Q is a quasigroup of order n, it is displayed as `<quasigroup of order n>` in `LOOPS`. Similarly, a loop of order n appears as `<loop of order n>`.
The displayed information for a loop L is enhanced when it is known that L has certain additional properties. At this point, we support:

`< `associative` `loop` `.``.``.` >` ,

`< `extra` `loop` `.``.``.` >` ,

`< `Moufang` `loop` `.``.``.` >` ,

`< `C` `loop` `.``.``.` >` ,

`< `left` `Bol` `loop` `.``.``.` >` ,

`< `right` `Bol` `loop` `.``.``.` >` ,

`< `LC` `loop` `.``.``.` >` ,

`< `RC` `loop` `.``.``.` >` ,

`< `alternative` `loop` `.``.``.` >` ,

`< `left` `alternative` `loop` `.``.``.` >` ,

`< `right` `alternative` `loop` `.``.``.` >` ,

`< `flexible` `loop` `.``.``.` >` .

The corresponding mathematical definitions and an example can be found in subsection 2.10.4.
It is possible for a loop to have several of the above properties. In such a case, we display the first property on the list that is satisfied. For instance, a left alternative flexible loop will appear as `<left alternative loop ...>`.
By default, the mth element of a quasigroup appears as `qm`, the mth element of a loop appears as `lm`, and the neutral element of a loop is denoted by `l1`. However, it is possible to change the names of elements of a quasigroup Q or loop L to name `m` with
· `SetQuasigroupElmName``( Q, name ) F`
· `SetLoopElmName``( L, name ) F`
Also see the example in Section 2.8.

### 2.1.3 Print mode

Elements of quasigroups and loops appear in the same way in both `View` and `Print` modes.
For quasigroups and loops in the `Print` mode, we display the multiplication table (if it is known), or we display the elements. See Section 2.4 for another way in which multiplication tables can be displayed.
In the following example, L is a loop with two elements.

    gap> L;
    <loop of order 2>
    gap> Print( L );
    <loop with multiplication table
    [ [  1,  2 ],
      [  2,  1 ] ]
    >
    gap> Elements( L );
    [ l1, l2 ]
    gap> SetLoopElmName( L, "loop_element" );; Elements( L );
    [ loop_element1, loop_element2 ]

## 2.2 Creating quasigroups and loops

As mentioned above, quasigroups and loops are represented by multiplication tables, which we also refer to as Cayley tables . When Q is a quasigroup of order n, the associated Cayley table is an n×n array with symbols 1, ..., n such that `qi`·`qj` = `qk` if and only if the entry in row i and column j is k. Similarly for loops.
The Cayley table can be entered manually, or read off from a file.

### 2.2.1 Testing multiplication tables

The following synonymous operations test if a multiplication table is a multiplication table of a quasigroup, i.e. a Latin squares with symbols 1, ..., n.
· `IsQuasigroupTable``( L ) A`
· `IsQuasigroupCayleyTable``( L ) A`
A Latin square on 1, ..., n is said to be normalized if the first column and the first row read 1, ..., n. Cayley table of a loop is therefore just another name for a normalized Latin square. The following operations test if L is a normalized Latin square:
· `IsLoopTable``( L ) A`
· `IsLoopCayleyTable``( L ) A`
A Latin square can be normalized by permuting its columns so that the first row reads 1, ..., n, and then permuting its rows so that the first column reads 1, ..., n. Note that it matters whether rows or columns are permuted first (see subsection 2.2.4 for more). This normalization is achieved in `LOOPS` with
· `NormalizedQuasigroupTable``( L ) F`
We would like to call the attention to the fact that the package `GUAVA` also has some operations dealing with Latin squares (in particular, the function `IsLatinSquare` is defined in `GUAVA`).

### 2.2.2 Creating quasigroups and loops manually

When L is a Latin square on 1, ..., n, the corresponding quasigroup is obtained with
· `QuasigroupByCayleyTable``( L ) F`
· `QuasigroupByCayleyTable``( L, <name> ) F`
If `<name>` is given, then the output of the quasigroup elements will have the form `<name>1`, `<name>2`, ...
When L is normalized, the corresponding loop is returned by
· `LoopByCayleyTable``( L ) F`

### 2.2.3 Creating quasigroups and loops from a file

Typing a large multiplication table manually is tedious and error-prone. We have therefore included a universal algorithm in `LOOPS` that reads multiplication tables of quasigroups from a file. Instead of writing a separate algorithm for each common format, our algorithm relies on the user to provide a bit of information about the input file. Here is an outline of the algorithm:

> Input: filename F, string D
> Step 1: Read the entire content of F into a string S.
> Step 2: Replace all end-of-line characters in S by spaces.
> Step 3: Replace by spaces all characters of S that appear in D.
> Step 4: Split S into maximal substrings without spaces, called chunks.
> Step 5: Recognize distinct chunks. Let n be the number of distinct chunks.
> Step 6: If the number of chunks is not n2, report error.
> Step 7: Construct the multiplication table by assigning numerical values 1, ..., n to chunks, depending on their position among distinct chunks.

The following examples clarify the algorithm and document its versatility:

input file

string D

resulting mult. table

comments

0 1 2 1

2 0 2

0 1

""

1 2 3

2 3 1

3 1 2

Data does not have to be

arranged into an array of any kind.

red green

green red

""

1 2

2 1

Chunks can be any strings.

[ [0, 1], [1, 0] ]

"[,]"

1 2

2 1

A typical table produced by ``GAP``

is easily parsed by deleting

brackets and commas.

x&y&z\hline

y&z&x\hline

z&x&y

"&\\einlh"

1 2 3

2 3 1

3 1 2

A typical TEX table with rows

separated by lines. We must use

"\\" in D because "\\" represents

the string "\" in ``GAP``.

I am as mad as

I say I am

"day"

1 2 3

2 3 1

3 1 2

Just for fun.

And here are the needed `LOOPS` commands:
· `QuasigroupFromFile``( F, D ) F`
· `LoopFromFile``( F, D ) F`

### 2.2.4 Conversions

We provide conversion operations that convert between magmas, quasigroups, loops and groups, provided such conversions are possible.
If M is a declared magma that happens to be a quasigroup, the corresponding quasigroup is returned via
· `AsQuasigroup``( M ) F`
If M is a magma that happens to be a quasigroup, the operation
· `AsLoop``( M ) F`
returns a loop L as follows:
If M possesses a neutral element e and f is the first element of M, then L is an isomorphic copy of M via the transposition (e,f).
If M does not posses a neutral element, then L is returned as

``PrincipalLoopIsotope``(` `M``,` `M``.``1``,` `M``.``1` `)``,

where
· `PrincipalLoopIsotope``( Q, f, g ) F`
is the principal isotope of Q using elements f, g of Q, as explained in Subsection 1.4.3.
Of course, one can obtain a loop from M in different ways, for instance by normalizing the Cayley table of M. The following example shows that these three approaches can yield different results in general:

    gap> A := [[2,1,5,3,4],[1,2,3,4,5],[5,3,4,1,2],[3,4,2,5,1],[4,5,1,2,3]];;
    gap> Q := QuasigroupByCayleyTable( A );;
    gap> L := AsLoop( Q );;
    gap> C := LoopByCayleyTable( NormalizedQuasigroupTable( CayleyTable(L) ) );;
    gap> P := PrincipalLoopIsotope( Q, Elements( Q )[ 2 ], Elements( Q )[ 2 ] );
    gap> CayleyTable( L );
    [ [ 1, 2, 3, 4, 5 ], [ 2, 1, 5, 3, 4 ], [ 3, 5, 4, 2, 1 ],
      [ 4, 3, 1, 5, 2 ], [ 5, 4, 2, 1, 3 ] ]
    gap> CayleyTable( C );
    [ [ 1, 2, 3, 4, 5 ], [ 2, 1, 4, 5, 3 ], [ 3, 5, 1, 2, 4 ],
      [ 4, 3, 5, 1, 2 ], [ 5, 4, 2, 3, 1 ] ]
    gap> CayleyTable( P );
    [ [ 1, 2, 3, 4, 5 ], [ 2, 1, 4, 5, 3 ], [ 3, 4, 5, 2, 1 ],
      [ 4, 5, 1, 3, 2 ], [ 5, 3, 2, 1, 4 ] ]

Finally, when M is a declared magma that happens to be a group, then the corresponding group is returned by
· `AsGroup``( M ) F`
Note that the conversions work in both directions, not just toward more special structures. Thus, if G is a declared group, then `AsLoop( G )` returns the corresponding loop, for instance.

### 2.2.5 Products of loops

Let L1, ..., Ln be a list consisting of loops and groups, where n ³ 1. Then
· `DirectProduct``( L1, ..., Ln ) F`
returns the direct product of L1, ..., Ln.
If there are only groups on the list, a group is returned, otherwise a loop is returned. If n=1, L1 is returned.

### 2.2.6 Opposite quasigroups and loops

When Q is a quasigroup with multiplication ·, the opposite quasigroup of Q is a quasigroup with the same underlying set as Q and with multiplication * defined by x*y=y·x.
Since the quasigroup-theoretical concepts are often oriented (cf. left Bol loops versus right Bol loops), it is useful to have access to the opposite quasigroup of Q:
· `Opposite``( Q ) F`

## 2.3 GAP categories

One can test if an element x belong to a quasigroup or to a loop, or if a given object Q is a quasigroup or a loop:
· `IsQuasigroupElement``( x ) category`
· `IsLoopElement``( x ) category`
· `IsQuasigroup``( Q ) category`
· `IsLoop``( Q ) category`

## 2.4 Basic attributes

The list of elements of a quasigroup Q is obtained by the usual command
· `Elements``( Q ) A`
The Cayley table of a quasigroup Q is returned with
· `CayleyTable``( Q ) A`
One can use `Display( CayleyTable( Q ) )` for pretty matrix-style output of small Cayley tables.
The neutral element of a loop L is obtained via
· `One``( L ) A`
If you want to know if a quasigroup Q has a neutral element, you can find out with the standard function for magmas
· `MultiplicativeNeutralElement``( Q ) A`
The size of a quasigroup Q is calculated by
· `Size``( Q ) A`
When L is a power associative loop (i.e., the orders of elements are well-defined in L), the exponent of L is the smallest positive integer divided by orders of all elements of L. The following attribute calculates the exponent without testing for power associativity:
· `Exponent``( L ) A`

    gap> Q := QuasigroupByCayleyTable( [ [ 1, 2 ], [ 2, 1 ] ] );
    <quasigroup of order 2>
    gap> [ IsQuasigroup( Q ); IsLoop( Q ); Size( Q ); Elements( Q ); ]
    [ true, false, 2, [ q1, q2 ] ]
    gap> IsQuasigroupElement( Elements( Q )[ 2 ] );
    true
    gap> CayleyTable( Q );
    [ [ 1, 2 ], [ 2, 1 ] ]

## 2.5 Basic arithmetic operations

Each quasigroup element in `GAP` knows which quasigroup it belongs to. It is therefore possible to perform arithmetic operations with quasigroup elements without referring to the quasigroup. All elements involved in the calculation must belong to the same quasigroup.

### 2.5.1 Multiplication

Two elements x, y of the same quasigroup are multiplied by x*y in `GAP`. Since multiplication of elements is ambiguous in the nonassociative case, we always multiply element from left to right (i.e., x*y*z means (x*y)*z)). Of course, one can specify association by parentheses.

### 2.5.2 Division

Universal algebraists introduce two additional operations for quasigroups. Namely the left division x\y satisfying x·(x\y)=y, and the right division x/y satisfying x/y·y=x. Since the two symbols / and \ are already used for different purposes in `GAP`, we instead introduce:
· `LeftDivision``( x, y ) O`
· `RightDivision``( x, y ) O`

### 2.5.3 Powers and inverses

Powers of elements are not well-defined in quasigroups, since bracketing can matter even for a single element. We say that the quasigroup Q is monoassociative , if for any x Î Q, the submagma generated by x is associative. Similarly, the loop L is said to be power associative , if for any x Î L, the subloop generated by x is associative.
For magmas and positive integer exponents, `GAP` defines the powers in the following way: x1=x, x2k=(xk)·(xk) and x2k+1=(x2k)·x for positive integer k. One can easily see that this returns xk in log2(k) steps. For `LOOPS`, we decided to keep this method, hoping that everybody will use it with care for quasigroups.
Let x be an element of a loop L with neutral element 1. Then the left inverse x¢ of x is the unique element of L satisfying x¢x=1. Similarly, the right inverse x" satisfies xx"=1. If x¢=x", we call x-1=x¢=x" the inverse of x.
· `LeftInverse``( x ) O`
· `RightInverse``( x ) O`
· `Inverse``( x ) O`
The following examples illustrates the usage of arithmetic operations. `MoufangLoop` will be explained in Chapter 4. In this example, `M.i` coincides with `Elements( M )[ i ]`.

    gap> M := MoufangLoop( 12, 1 );; x := M.2;
    l2
    gap> [ x * M.3, x^2, x^(-1), Inverse( x ) ];
    [ l4, l1, l2, l2 ]
    gap> One( M ) = LeftDivision( x, x );
    true

### 2.5.4 Associators and commutators

Let Q be a quasigroup and x, y, z Î Q. Then the associator of x, y, z is the unique element u such that (xy)z=(x(yz))u. The commutator of x, y is the unique element v such that xy=(yx)v.
· `Associator``( x, y, z ) O`
· `Commutator``( x, y ) O`

## 2.6 Generators

The following two attributes are synonyms of `GeneratorsOfMagma`.
· `GeneratorsOfQuasigroup``( Q ) A`
· `GeneratorsOfLoop``( L ) A`
As usual in `GAP`, one can refer to the ith generator of a quasigroup Q by `Q.i`. Note that it is not necessarily the case that `Q.i = Elements( Q )[ i ]`, since the set of generators can be a proper subset of the elements.
It is easy to prove that a loop of order n can be generated by a subset containing at most log2n elements. Such a set is returned via
· `SmallGeneratingSet``( L ) A`

## 2.7 Permutations associated with loops

### 2.7.1 Sections

The following two attributes calculate the left and right section of a quasigroup Q:
· `LeftSection``( Q ) A`
· `RightSection``( Q ) A`
Given an element x of a quasigroup Q, the left and right translations of Q by x are obtained by
· `LeftTranslation``( Q, x ) F`
· `RightTranslation``( Q, x ) F`

### 2.7.2 Multiplication groups

The left multiplication group, right multiplication group and the multiplication group of a quasigroup Q is calculated as follows:
· `LeftMultiplicationGroup``( Q ) A`
· `RightMultiplicationGroup``( Q ) A`
· `MultiplicationGroup``( Q ) A`
The relative versions of multiplication groups are implemented only for loops. When S is a subloop of a loop L, the following functions return the relative multiplication groups:
· `RelativeLeftMultiplicationGroup``( L, S ) F`
· `RelativeRightMultiplicationGroup``( L, S ) F`
· `RelativeMultiplicationGroup``( L, S ) F`

### 2.7.3 Inner mapping group

The inner mapping group of a loop L is the stabilizer of the unit element in Mlt(L). The elements of this stabilizer are called inner maps of L. The inner mapping group of a loop L is obtained by:
· `InnerMappingGroup``( L ) A`

    gap> M := MoufangLoop( 12, 1 );
    <Moufang loop 12/1>
    gap> LeftSection( M )[ 2 ];
    (1,2)(3,4)(5,6)(7,8)(9,12)(10,11)
    gap> Mlt := MultiplicationGroup( M ); Inn := InnerMappingGroup( M );
    <permutation group of size 2592 with 23 generators>
    Group([ (4,6)(7,11), (7,11)(8,10), (2,6,4)(7,9,11), (3,5)(9,11), (8,12,10) ])
    gap> Size( Inn );
    216

## 2.8 Subquasigroups and subloops

Let Q be a quasigroup and S a subquasigroup of Q. Since the multiplication in S coincides with the multiplication in Q, it is reasonable not to store the multiplication table of S. However, the quasigroup S then must know that it is a subquasigroup of Q. In order to facilitate this relationship, we introduce the attribute
· `Parent``( Q ) A`
for quasigroups. When Q is not created as a subquasigroup of another quasigroup, the attribute `Parent( Q )` is set to Q. When Q is created as a subquasigroup of a quasigroup H, we set `Parent( Q ) := Parent( H )`. Thus, in effect, `Parent( Q )` is the largest quasigroup from which Q was created.
Given a collection C of elements of a quasigroup Q, the function
· `PosInParent``( C ) F`
returns the list of positions of the elements of C among the elements of `Parent( Q )`. The multiplication in Q can therefore be easily reconstructed from `Parent( Q )` via `PosInParent`.
When S is a subset of a quasigroup Q (loop L), the subquasigroup of Q (subloop of L) is returned via:
· `Subquasigroup``( Q, S ) F`
· `Subloop``( L, S ) F`
Finally, the following two functions test if a quasigroup (loop) S is a subquasigroup (subloop) of a quasigroup Q (loop L). Note that the functions return false when S and Q (L) do not have the same parent.
· `IsSubquasigroup``( Q, S ) F`
· `IsSubloop``( L, S ) F`
The following example illustrates the main features of the subquasigroup construction. Note how the Cayley table of the subquasigroup is created only upon explicit demand. Also note that changing the names of elements of a suquasigroup (subloop) automatically changes the names of the elements of the parent subquasigroup (subloop). This is because the elements are shared.

    gap> M := MoufangLoop( 12, 1 );; S := Subloop( M, [ M.5 ] );
    <loop of order 3>
    gap> [ Parent( S ) = M, Elements( S ), PosInParent( S ) ];
    [ true, [ l1, l3, l5], [ 1, 3, 5 ] ]
    gap> HasCayleyTable( S );
    false
    gap> SetLoopElmName( S, "s" );; Elements( S ); Elements( M );
    [ s1, s3, s5 ]
    [ s1, s2, s3, s4, s5, s6, s7, s8, s9, s10, s11, s12 ]
    gap> CayleyTable( S );
    [ [ 1, 2, 3 ], [ 2, 3, 1 ], [ 3, 1, 2 ] ]
    gap> [ HasCayleyTable( S ), Parent( S ) = M ];
    [ true, true]
    gap> L := LoopByCayleyTable( CayleyTable( S ) );
    <loop of order 3>
    gap> [ Parent( L ) = L, IsSubloop( M, S ), IsSubloop( M, L ) ];
    [ true, true, false ]

## 2.9 Nucleus, commutant, center

Let Q be a quasigroup. The left nucleus Nl(Q) of Q is the set {x Î Q; x(yz)=(xy)z for every y, z Î Q}. One defines similarly the middle nucleus Nm(Q) and the right nucleus Nr(Q). Then the nucleus N(Q) of Q is equal to Nl(Q)ÇNm(Q)ÇNr(Q). These nuclei are calculated in `LOOPS` as follows:
· `LeftNucleus``( Q ) A`
· `MiddleNucleus``( Q ) A`
· `RightNucleus``( Q ) A`
· `Nuc``( Q ) A`
The word `Nucleus` is reserved in the core `GAP` system for a function in two variables, therefore we use the abbreviation `Nuc`, which is also common in the literature. However, we support these synonyms of `Nuc`:
· `NucleusOfLoop``( Q ) A`
· `NucleusOfQuasigroup``( Q ) A`
Since all nuclei are subquasigroups of Q, they are returned as subquasigroups (or subloops).
The commutant C(Q) of Q is the set {x Î Q; xy=yx for every y Î Q}. It is also known under the name Moufang center . It is obtained via
· `Commutant``( Q ) A`
The center Z(Q) is defined as C(Q)ÇN(Q), and it is obtained via
· `Center``( Q ) A`
Finally, the associator subloop of a loop L is the subloop of L generated by all associators of L. (Note that some authors define the associator subloop as the smallest normal subloop A of L such that L/A is associative. The two definitions are not equivalent in general.)
· `AssociatorSubloop``( L ) A`

## 2.10 Testing properties

The reader should be aware that although loops are quasigroups, it is often the case in the literature that a property named P can differ for quasigroups and loops; for instance, a Steiner loop is not necessarily a Steiner quasigroup. To avoid such ambivalences, we often include the noun `Loop` or `Quasigroup` as part of the name of the property; e.g. `IsSteinerQuasigroup` versus `IsSteinerLoop`. On the other hand, some properties coincide for quasigroups and loops and we therefore do not include `Loop`, `Quasigroup` as part of the name of the property; e.g. `IsCommutative`.

### 2.10.1 Associativity, commutativity and generalizations

The following properties test if a quasigroup Q is associative and commutative.
· `IsAssociative``( Q ) P`
· `IsCommutative``( Q ) P`
A loop L is said to be power associative (resp. diassociative) if every monogenic subloop of L (resp. every 2-generated subloop of L) is a group.
· `IsPowerAssociative``( L ) P`
· `IsDiassociative``( L ) P`

### 2.10.2 Inverse properties

A loop L has the left inverse property if x¢(xy)=y for every x, y Î L, where x¢ is the left inverse of x. Dually, L has the right inverse property if (yx)x"=y for every x, y Î L, where x" is the right inverse of x. If L has both the left and right inverse property, it has the inverse property . We say that L has two-sided inverses if x¢=x" for every x Î L.
· `HasLeftInverseProperty``( L ) P`
· `HasRightInverseProperty``( L ) P`
· `HasInverseProperty``( L ) P`
· `HasTwosidedInverses``( L ) P`
A loop L with two-sided inverses has the automorphic inverse property if (xy)-1=x-1y-1 for every x, y Î L. Similarly, it has the antiautomorphic inverse property if (xy)-1=y-1x-1.
· `HasAutomorphicInverseProperty``( L ) P`
· `HasAntiautomorphicInverseProperty``( L ) P`
There are many additional inverse properties but we decided against including them at this stage.

### 2.10.3 Properties of quasigroups

A quasigroup Q is semisymmetric if (xy)x=y for every x, y Î Q. Equivalently, Q is semisymmetric if x(yx)=y for every x, y Î Q. A semisymmetric commutative quasigroup is known as totally symmetric . Totally symmetric quasigroups are precisely quasigroups satisfying xy=x\y = x/y.
· `IsSemisymmetric``( Q ) P`
· `IsTotallySymmetric``( Q ) P`
A quasigroup Q is idempotent if x2=x for every x Î Q. Idempotent totally symmetric quasigroups are known as Steiner quasigroups . A quasigroup Q is unipotent if x2=y2 for every x, y Î Q.
· `IsIdempotent``( Q ) P`
· `IsSteinerQuasigroup``( Q ) P`
· `IsUnipotent``( Q ) P`
A quasigroup is left distributive if it satisfies x(yz)=(xy)(xz). Similarly, it is right distributive if it satisfies (xy)z=(xz)(yz). A distributive quasigroup is a quasigroup that is both left and right distributive. A quasigroup is called entropic or medial if it satisfies (xy)(zw)=(xz)(yw).
· `IsLeftDistributive``( Q ) P`
· `IsRightDistributive``( Q ) P`
· `IsDistributive``( Q ) P`
· `IsEntropic``( Q ) P`
· `IsMedial``( Q ) P`
TO be compatible with GAP's terminology, we also support the synonyms
· `IsLDistributive``( Q ) P`
· `IsRDistributive``( Q ) P`
for `IsLeftDistributive` and `IsRightDistributive`, respectively.

### 2.10.4 Loops of Bol-Moufang type and related properties

Following Fenyves, [1969] and Phillips and Vojtechovský, [2004], a variety of loops is said to be of Bol-Moufang type if it is defined by a single identity of Bol-Moufang type, i.e., by an identity that: (i) contains the same 3 variables on both sides, (ii) exactly one of the variables occurs twice on both sides, (iii) the variables occur in the same order on both sides.
It is proved in Phillips and Vojtechovský, [2004] that there are 13 varieties of nonassociative loops of Bol-Moufang type, as summarized in Figure 2.1.
Picture Omitted
Figure 2.1: Varieties of loops of Bol-Moufang type.
Note that although some of the defining identities are not of Bol-Moufang type, they are equivalent to a Bol-Moufang identity. Moreover, some varieties in the Figure are defined by several, equivalent identities of Bol-Moufang type.
There are several varieties related to loops of Bol-Moufang type. A loop is said to be: alternative if it is both left and right alternative; nuclear square loop if it is left, middle and right nuclear square.
Here are the corresponding `GAP` commands (argument L indicates that the property applies only to loops, argument Q indicates that the property applies to quasigroups):
· `IsExtraLoop``( L ) P`
· `IsMoufangLoop``( L ) P`
· `IsCLoop``( L ) P`
· `IsLeftBolLoop``( L ) P`
· `IsRightBolLoop``( L ) P`
· `IsLCLoop``( L ) P`
· `IsRCLoop``( L ) P`
· `IsLeftNuclearSquareLoop``( L ) P`
· `IsMiddleNuclearSquareLoop``( L ) P`
· `IsRightNuclearSquareLoop``( L ) P`
· `IsNuclearSquareLoop``( L ) P`
· `IsFlexible``( Q ) P`
· `IsLeftAlternative``( Q ) P`
· `IsRightAlternative``( Q ) P`
· `IsAlternative``( Q ) P`
All inclusions among the varieties of loops of Bol-Moufang type are summarized in Figure 2.1, and all these inclusions are built into the `LOOPS` package. (See Section 2.15.)
The following trivial example shows some of the implications and the naming conventions of `LOOPS` at work:

    gap> L := LoopByCayleyTable( [ [ 1, 2 ], [ 2, 1 ] ] );
    <loop of order 2>
    gap> IsLeftBolLoop( L );
    true
    gap> [ HasIsLeftAlternativeLoop( L ), IsLeftAlternativeLoop( L ) ];
    [ true, true ]
    gap> [ HasIsRightBolLoop( L ), IsRightBolLoop( L ) ];
    [ false, true ]
    gap> L;
    <Moufang loop of order 2>
    gap> [ IsAssociative( L ), L ];
    [ true, <associative loop of order 2> ]

The analogous terminology for quasigroups of Bol-Moufang type is not standard yet, and hence is not supported in `LOOPS`.

### 2.10.5 Conjugacy closed loops

A loop is left (resp. right) conjugacy closed if its left (resp. right) translations are closed under composition. A loop that is both left and right conjugacy closed is called conjugacy closed . It is common to refer to these loops are LCC, RCC, CC-loops, respectively.
· `IsLCCLoop``( L ) P`
· `IsRCCLoop``( L ) P`
· `IsCCLoop``( L ) P`
The equivalence LCC + RCC = CC is built into the `LOOPS` package.

### 2.10.6 Additional varieties of loops

A left (resp. right) Bol loop with the automorphic inverse property is known as left (resp. right) Bruck loop . Bruck loops are also known as K-loops .
· `IsLeftBruckLoop``( L ) P`
· `IsLeftKLoop``( L ) P`
· `IsRightBruckLoop``( L ) P`
· `IsRightKLoop``( L ) P`
Steiner loop is an inverse property loop of exponent 2.
· `IsSteinerLoop``( L ) P`

## 2.11 Normality

A subloop S of a loop L is normal if it is invariant under all inner mappings of L. Normality is tested via:
· `IsNormal``( L, S ) F`
When S is a subset of a loop L or a subloop of L the normal closure of S in L is the smallest normal subloop of L containing S. It is obtained by
· `NormalClosure``( L, S ) F`
A loop L is simple if all normal subloops of L are trivial. The corresponding test in `LOOPS` is:
· `IsSimple``( L ) P`

## 2.12 Factor loop

When N is a normal subloop of a loop L, the factor loop L/N can be obtained directly via the command `L/N`, or by
· `FactorLoop``( L, N ) F`
The natural projection from L to L/N is returned as follows:
· `NaturalHomomorphismByNormalSubloop``( L, N ) F`

    gap> M := MoufangLoop( 12, 1 );; S := Subloop( M, [ M.3 ] );
    <loop of order 3>
    gap> IsNormal( M, S );
    true
    gap> F := FactorLoop( M, S );
    <loop of order 4>
    gap> NaturalHomomorphismByNormalSubloop( M, S );
    MappingByFunction( <loop of order 12>, <loop of order 4>,
        function( elm ) ... end )

## 2.13 Nilpotency

The definition of nilpotency and nilpotence class is the same as in group theory. The corresponding commands are:
· `NilpotencyClassOfLoop``( L ) A`
· `IsNilpotent``( L ) P`
When L is not nilpotent, `NilpotencyClassOfLoop( L )` returns `fail`.
A loop L is said to be strongly nilpotent if its multiplication group is nilpotent. This property is obtained by:
· `IsStronglyNilpotent``( L ) P`

## 2.14 Solvability

The definition of solvability, derived subloop, derived length, Frattini subloop and Frattini factor size is the same as for groups. Frattini subloop is calculated only for strongly nilpotent loops.
· `IsSolvable``( L ) P`
· `DerivedSubloop``( L ) A`
· `DerivedLength``( L ) A`
· `FrattiniSubloop``( L ) A`
· `FrattinifactorSize``( L ) A`

## 2.15 Filters built into `LOOPS`

Many implications among properties of loops are built directly into `LOOPS`. A sizeable portion of these properties are of trivial character or are based on definitions (e.g., alternative loops = left alternative loops + right alternative loops). The remaining implications are theorems.
All filters of `LOOPS` are summarized below (using the `GAP` convention that the property on the left is implied by the property (properties) on the right).

    ( IsExtraLoop,  IsAssociative and IsLoop )
    ( IsDiassociative, IsAssociative and IsLoop )
    ( HasInverseProperty, HasRightInverseProperty and IsCommutative )
    ( HasInverseProperty, HasLeftInverseProperty and IsCommutative )
    ( IsMoufangLoop, IsRightBolLoop and IsCommutative )
    ( IsMoufangLoop, IsLeftBolLoop and IsCommutative )
    ( IsMoufangLoop, IsRightBruckLoop and IsCommutative )
    ( IsMoufangLoop, IsLeftBruckLoop and IsCommutative )
    ( IsRightNuclearSquareLoop, IsLeftNuclearSquareLoop and IsCommutative )
    ( IsLeftNuclearSquareLoop, IsRightNuclearSquareLoop and IsCommutative )
    ( HasAutomorphicInverseProperty, HasAntiautomorphicInverseProperty and IsCommutative )
    ( HasAntiautomorphicInverseProperty, HasAutomorphicInverseProperty and IsCommutative )
    ( IsAlternative, IsLeftAlternative and IsCommutative )
    ( IsAlternative, IsRightAlternative and IsCommutative )
    ( HasTwosidedInverses, IsPowerAssociative )
    ( IsPowerAssociative, IsDiassociative )
    ( IsAlternative, IsDiassociative )
    ( IsFlexible, IsDiassociative )
    ( HasLeftInverseProperty, HasInverseProperty )
    ( HasRightInverseProperty, HasInverseProperty )
    ( HasTwosidedInverses, HasInverseProperty )
    ( HasInverseProperty, HasLeftInverseProperty and HasRightInverseProperty )
    ( IsMoufangLoop, IsExtraLoop )
    ( IsNuclearSquareLoop, IsExtraLoop )
    ( IsCLoop, IsExtraLoop )
    ( IsExtraLoop, IsMoufangLoop and IsLeftNuclearSquareLoop )
    ( IsExtraLoop, IsMoufangLoop and IsMiddleNuclearSquareLoop )
    ( IsExtraLoop, IsMoufangLoop and IsRightNuclearSquareLoop )
    ( IsLeftBolLoop, IsMoufangLoop )
    ( IsRightBolLoop, IsMoufangLoop )
    ( IsFlexible, IsMoufangLoop )
    ( IsDiassociative, IsMoufangLoop )
    ( IsMoufangLoop, IsLeftBolLoop and IsRightBolLoop )
    ( IsLCLoop, IsCLoop )
    ( IsRCLoop, IsCLoop )
    ( IsCLoop, IsLCLoop and IsRCLoop )
    ( IsLeftAlternative, IsLeftBolLoop )
    ( HasTwosidedInverses, IsLeftBolLoop )
    ( IsRightAlternative, IsRightBolLoop )
    ( HasTwosidedInverses, IsRightBolLoop )
    ( IsLeftAlternative, IsLCLoop )
    ( IsLeftNuclearSquareLoop, IsLCLoop )
    ( IsMiddleNuclearSquareLoop, IsLCLoop )
    ( IsPowerAssociative, IsLCLoop )
    ( IsRightAlternative, IsRCLoop )
    ( IsRightNuclearSquareLoop, IsRCLoop )
    ( IsMiddleNuclearSquareLoop, IsRCLoop )
    ( IsPowerAssociative, IsRCLoop )
    ( IsLeftNuclearSquareLoop, IsNuclearSquareLoop )
    ( IsRightNuclearSquareLoop, IsNuclearSquareLoop )
    ( IsMiddleNuclearSquareLoop, IsNuclearSquareLoop )
    ( IsNuclearSquareLoop, IsLeftNuclearSquareLoop and IsRightNuclearSquareLoop and IsMiddleNuclearSquareLoop )
    ( IsLeftAlternative, IsAlternative )
    ( IsRightAlternative, IsAlternative )
    ( IsAlternative, IsLeftAlternative and IsRightAlternative )
    ( IsLCCLoop, IsCCLoop )
    ( IsRCCLoop, IsCCLoop )
    ( IsCCLoop, IsLCCLoop and IsRCCLoop )
    ( HasAutomorphicInverseProperty, IsLeftBruckLoop )
    ( IsLeftBolLoop, IsLeftBruckLoop )
    ( IsLeftBruckLoop, IsLeftBolLoop and HasAutomorphicInverseProperty )
    ( HasAutomorphicInverseProperty, IsRightBruckLoop )
    ( IsRightBolLoop, IsRightBruckLoop )
    ( IsRightBruckLoop, IsRightBolLoop and HasAutomorphicInverseProperty )
    ( IsCommutative, IsSteinerLoop )
    ( HasInverseProperty, IsSteinerLoop )

# Chapter 3
Specific methods

## 3.1 Isomorphisms and automorphisms

All isomorphisms between two loops can be found with `LOOPS`. The function
· `IsomorphismLoops``( L, M ) F`
returns a single isomorphism between loops L, M, if the loops are isomorphic, and it fails otherwise. If an isomorphism exists it is returned as a permutation p Î S|L|, where p(i)=j means that the ith element of L is mapped onto the jth element of M.
The function
· `AutomorphismGroup``( L ) F`
returns the automorphism group of the loop L. Since two isomorphisms differ by an automorphism, all isomorphisms can be obtained by the above two functions.

### 3.1.1 Discriminator

In order to speed up the search for isomorphisms and automorphisms, we first calculate some loop invariants preserved under isomorphisms, and use these invariants to partition the loop into blocks of elements preserved under isomorphism. These invariants for a loop L can be obtained via
· `Discriminator`( L ) F
Since the details are technical, we will not present them here. See Vojtechovský, [2004] for more.
If two loops have different discriminators, they are not isomorphic. If they have identical discriminator, they may or may not be isomorphic. The function
· `AreEqualDiscriminators`( D1, D2 ) F
returns true if the discriminators D1, D2 are equal.
Given a loop L and its discriminator D, the function
· `EfficientGenerators`( L, D ) F
returns a generating set of L that is optimized with respect to the discriminator D. Once again, the details are too technical to be presented here. The returned set of generators is usually very small. Also see `SmallGeneratingSet.

## 3.2 Moufang modifications
Ales Drápal discovered two prominent families of extensions of Moufang loops. These extensions can be used to obtain many, perhaps all, nonassociative Moufang loops of order at most 64. We call these two constructions Moufang modifications . The library of Moufang loops included with `LOOPS is based on Moufang modifications. We describe the two modifications briefly here. See Drápal and Vojtechovský, [2004] for details.

### 3.2.1 Cyclic modification
Assume that L is a Moufang loop with normal subloop S such that L/S is a cyclic group of order 2m. Let h Î SÇZ(L). Let a be a generator of L/S and write L = Èi Î M ai, where M={-m+1, ..., m}. Let s:\mathbb Z® M be defined by

s

(i) = ì
ï
í
ï
î

0,

i Î

M,

1,

i > m,

-

1,

i < -m+1.

Introduce a new multiplication * on L defined by

x*y = xyhs

(i+j), where x Î ai, y Î aj, i Î M, j Î M. Then (L,*) is a Moufang loop, a cyclic modification of L.
When L, S, a, h are as above and when a is any element of a, the corresponding cyclic modification is obtained via
· `LoopByCyclicModification``( L, S, a, h ) F`

### 3.2.2 Dihedral modification

Assume that L is a Moufang loop with normal subloop S such that L/S is a dihedral group of order 4m, with m ³ 1. Let M and s be defined as in the cyclic case. Let b, g Î L/S be two involutions of L/S such that a = bg generates a cyclic subgroup of L/S of order 2m. Let e Î b and f Î g be arbitrary. Then L can be written as a disjoint union L=Èi Î M(aiÈeai), and also L=Èi Î M(aiÈaif. Let G0=Èi Î Mai, and G1=L\G0. Let h Î SÇN(L)ÇZ(G0). Introduce a new multiplication * n L defined by

x*y = xyh(-

1)rs(i+j), where x Î aiÈeai, y Î ajÈajf, i Î M, j Î M, y Î Gr, r Î {0,1}. Then (L,*) is a Moufang loop, a dihedral modification of L.
When L, S, e, f and h are as above, the corresponding dihedral modification is obtained via
· `LoopByDihedralModification``( L, S, e, f, h ) F`

### 3.2.3 Loops M(G,2)

In order to apply the cyclic and dihedral modification, it is beneficial to have access to a class of nonassociative Moufang loops. The following construction is due to Chein:
Let G be a group. Let [`G]={[`g]; g Î G} be a set of new elements. Define multiplication * on L=GÈ[`G] by

g*h = gh, g*

h

=

hg

,

g

*h =

gh-

1
,

g
*

h
=h-1g, where g, h Î G. Then L=M(G,2) is a Moufang loop that is nonassociative if and only if G is nonabelian.
The loop M(G,2) can be obtained from a finite group G with
· `LoopMG2``( G ) F`
in `LOOPS`.

## 3.3 Triality for Moufang loops

Let G be a group and s, r be automorphisms of G, satisfying s2 = r3 = (sr)2 = 1. We write the automorphisms of a group as exponents and [g,s] for g-1gs. We say that the triple (G,r,s) is a group with triality if [g, s] [g,s]r [g,s]r2 = 1 holds for all g Î G. It is known that one can associate a group with triality (G,r,s) in a canonical way with a Moufang loop L. See Nagy and Vojtechovský, [2003] for more details.
For any Moufang loop L, we can calculate the triality group as a permutation group acting on 3|L| points. If the multiplication group of L is polycyclic, then we can also represent the triality group as a pc group. In both cases, the automorphisms s and r are in the same family as the elements of G.
Given a Moufang loop L, the function
· `TrialityPermGroup``( L ) F
returns a record [G, r, s], where G is the group with triality associated with L, and r, s are the corresponding triality automorphisms.
The function
· `TrialityPcGroup`( L ) F
differs from `TrialityPermGroup only in that G is returned as a pc group.

# Chapter 4
Libraries of small loops
Libraries of small loops are an integral part of `LOOPS.

## 4.1 A typical library
A library named "my Library" is stored in file `data/mylibrary.tbl, and the corresponding data structure is named `my_library_data.
The array `my_library_data consists of three lists: `my_library_data[ 1 ] is a list of orders for which there is at least one loop in the library, `my_library_data[ 2 ][ k ] is the number of loops of order `my_library_data[ 1 ][ k ] in the library, and `my_library_data[ 3 ][ s ] contains data necessary to produce the sth loop in the library. The format of `my_library_data[ 3 ] depends on the particular library and is not standardized in any way.
The user can retrieve the mth loop of order n from library named "my Library" according to the template
· `MyLibraryLoop`( n, m ) global function template
It is also possible to obtain the same loop with
· `LibraryLoop`( name, n, m ) F
where `name is the name of the library.
For example, when the library is called "left Bol", the corresponding data file is called `data/leftbol.tbl, the corresponding data structure is named `left_bol_data, and the mth left Bol loop of order n is obtained via
· `LeftBolLoop`( n, m ) F
or via
· `LibraryLoop`( "left Bol", n, m ) F
We are now going to describe the individual libraries in detail. A brief information about the library named `name can also be obtained in `LOOPS with
· `DisplayLibraryInfo`( name ) F

## 4.2 Left Bol loops
The library named "left Bol" contains all 6 nonassociative left Bol loops of order 8. Following the general pattern, the mth nonassociative left Bol loop of order n is obtained by
· `LeftBolLoop`( n, m ) F
We intend to enlarge this library significantly in future versions of `LOOPS, when the classification of small Bol loops is completed.

## 4.3 Small Moufang loops
The library named "Moufang" contains all nonassociative Moufang loops of order less than 64, and additional 4262 nonassociative Moufang loops of order 64. It is possible that there are no other nonassociative Moufang loops of order 64 than those contained in the library.
The mth nonassociative Moufang loop of order n is obtained by
· `MoufangLoop`( n, m ) F
For n £ 63, our catalog numbers coincide with those of Goodaire et al. Goodaire, May and Raman, [1999].
The extent of the library is summarized below:

order

12

16

20

24

28

32

36

40

42

44

48

52

54

56

60

64

loops in the libary

1

5

1

5

1

71

4

5

1

1

51

1

2

4

5

4262
The octonion loop of order 16 (i.e., the multiplication loop of the ± basis elements in the 8-dimensional standard real octonion algebra) is `MoufangLoop( 16, 3 )`.

### 4.3.1 Search for additional Moufang loops

Since we would like to know if there are additional nonassociative Moufang loops of order 64, we have implemented the function
· `IsomorphismTypeOfMoufangLoop`( L ) F
If L is a Moufang loop cataloged in `LOOPS as the mth Moufang loop of order n, the function returns `[[n,m],p], where p is a permutation of `[1..n] that is an isomorphism from L to the cataloged copy of L. If n=64 and L is Moufang loop not cataloged in `LOOPS, the user is prompted to contact the authors of `LOOPS.
In order to speed up the function `IsomorphismTypeOfMoufangLoop, we have precalculated and stored in the data file `data moufang_discriminators.tbl the discriminators of all Moufang loops in the library. The file is rather large (850 KB) and took about 20 minutes to precalculate. You can delete the file if you won't use `IsomorphismTypeOfMoufangLoop.

gap> D := DirectProduct( MoufangLoop( 16, 2 ), CyclicGroup( 2 ) ); <loop of order 32> gap> IsomorphismTypeOfMoufangLoop( D ); [ [ 32, 2 ], (2,3,12,20,11,29,23,13,30,31,28,27,22,15,32,18,10,19,16,24,14, 25,21,8,7,6,9,17,5) ] gap> A := AutomorphismGroup( D ); Size( A ); <permutation group with 34 generators> 3072

## 4.4 Steiner loops

Here is how the libary "Steiner" is described within `LOOPS`:

    gap> DisplayLibraryInfo( "Steiner" );
    The library contains all nonassociative Steiner loops of order less or equal
    to 16. It also contains the associative Steiner loops of order 4 and 8.
    ------
    Extent of the library:
       1 loop of order 4
       1 loop of order 8
       1 loop of order 10
       2 loops of order 14
       80 loops of order 16
    true

The mth Steiner loop of order n is obtained by
· `SteinerLoop``( n, m ) F
Our catalog numbers coincide with those of Colbourn and Rosa Colbourn and Rosa , [1999].

## 4.5 Paige loops
Paige loops are nonassociative finite simple Moufang loops. By Liebeck, [1987], there is precisely one Paige loop for every finite field GF(q).
The library named "Paige" contains the smallest nonassociative simple Moufang loop
· `PaigeLoop`( 2 ) F

## 4.6 Interesting loops
The library named ïnteresting" contains some loops that are illustrative for the theory of loops. At this point, the library contains a nonassociative loop of order 5, a nonassociative nilpotent loop of order 6, a nonMoufang left Bol loop of order 16, and the loop of sedenions of order 32 (sedenions generalize octonions).
The loops are obtained with
· `InterestingLoop`( n, m ) F

# Chapter 5
Plans for future versions
We hope that the `LOOPS package will become a standard computational tool in quasigroup theory and loop theory, and we therefore anticipate some interest among researchers in expanding the package. In this chapter, we present several possible directions in which this future expansion could lead. Since we will base our decision on your feedback, please let us know what you would like to see implemented in `LOOPS.

## 5.1 Alternative representations of quasigroups and loops in `GAP
(The word "representation" does not have the usual mathematical meaning in this section.) Direct products, semidirect products and many other constructions of loops can be represented in a more space-efficient way than by Cayley tables. Large Paige loops, generalizations of octonions and other loops can be represented nicely. None of these representations is currently implemented in `LOOPS.
Presentations of some loops within their varieties are known and perhaps should be found in `LOOPS.

## 5.2 Better support for quasigroups
This package is concerned primarily with loops. Although some functions are kept on a level general enough for quasigroups, many are not. Only a few quasigroup-theoretical properties are testable in `LOOPS at this point. The operations `LeftDivision and `RightDivision are awkward to work with.

## 5.3 More homomorphisms and homotopisms
The general concept of homomorphisms of quasigroup is obvious. Beside

``NaturalHomomorphismByNormalSubloop, ``PrincipalLoopIsotope, other homomorphisms and homotopisms should be defined for loops and quasigroups.

## 5.4 Expanded libraries
More Bol loops should be cataloged. Interesting loops should be gathered in a more systematic way. Loops could be cataloged not only up to isomorphism but also up to isotopism. We know how to create the library of all conjugacy closed loops of order p2. Any interest?

## 5.5 Bits and pieces
We would like to see the following features in a future version of `LOOPS: Mk laws, cross inverse property (CIP), weak inverse property (WIP), homotopisms.

## Bibliography

[Bruck 1971]
R. Hubert Bruck, A Survey of Binary Systems, third printing, corrected, Ergebnisse der Mathematik und ihrer Grenzgebiete, Neue Folge 20, Springer-Verlag, 1971.
[Chein 1990]
O. Chein, H. O. Pflugfelder, J. D. H. Smith (editors), Quasigroups and Loops: Theory and Applications, Sigma Series in Pure Mathematics 8, Heldermann Verlag Berlin, 1990.
[Colbourn and Rosa 1999]
Charles J. Colbourn and Alexander Rosa, Triple systems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1999.
[Drápal and Vojtechovský 2004]
Ales Drápal and Petr Vojtechovský, Moufang loops that share associator and three quarters of their multiplication tables, to appear in Rocky Mountain Journal of Mathematics.
[Fenyves 1969]
Ferenc Fenyves, Extra loops II, On loops with identities of Bol-Moufang type, Publ. Math. Debrecen 16(1969), 187-192.
[Goodaire, May and Raman 1999]
Edgar G. Goodaire, Sean May and Maitreyi Raman, The Moufang loops of order less than 64, Commack, NY: Nova Science Publishers, 1999.
[Liebeck 1987]
M. Liebeck, The classification of finite simple Moufang loops, Math. Proc. Cambridge Philos. Soc. 102 (1987), 33-47.
[Nagy and Vojtechovský 2003]
Gábor P. Nagy and Petr Vojtechovský, Octonions, simple Moufang loops and triality, Quasigroups and Related Systems 10 (2003), 65-94.
[Pflugfelder 1990]
Hala O. Pflugfelder, Quasigroups and Loops: Introduction, Sigma Series in Pure Mathematics 7, Heldermann Verlag Berlin, 1990.
[Phillips and Vojtechovský 2004]
J. D. Phillips and Petr Vojtechovský, Varieties of loops of Bol-Moufang type, submitted.
[Vojtechovský 2004]
Petr Vojtechovský, Toward the classification of Moufang loops of order 64, submitted.

## Index (showing section)

alternative loop, 2.10
antiautomorphic inverse property, 2.10
AreEqualDiscriminators, 3.1
AsGroup, 2.2
AsLoop, 1.5, 2.2
AsQuasigroup, 1.5, 2.2
Associator, 2.5
associator, 2.5
associator subloop, 2.9
AssociatorSubloop, 2.9
AutomorphismGroup, 3.1
Cayley table, 2.2
CayleyTable, 2.4
Center, 2.9
Commutant, 2.9
commutant, 2.9
Commutator, 2.5
commutator, 2.5
conjugacy closed loop, 2.10
cyclic modification, 3.2

DerivedLength, 2.14
DerivedSubloop, 2.14
diassoicativy, 2.10
dihedral modification, 3.2
DirectProduct, 2.2
Discriminator, 3.1
DisplayLibraryInfo, 4.1
distributive quasigroup, 2.10
EfficientGenerators, 3.1
Elements, 2.4
entropic quasigroup, 2.10
Exponent, 2.4
exponent, 2.4
FactorLoop, 2.12
FrattinifactorSize, 2.14
FrattiniSubloop, 2.14
GeneratorsOfLoop, 2.6
GeneratorsOfQuasigroup, 2.6
group, 1.4
groupoid, 1.4

HasAntiautomorphicInverseProperty, 2.10
HasAutomorphicInverseProperty, 2.10
HasInverseProperty, 2.10
HasLeftInverseProperty, 2.10
HasRightInverseProperty, 2.10
HasTwosidedInverses, 2.10
homomorphism, 1.4
homotopism, 1.4
idempotent, 2.10
identity element, 1.4
inner mapping group, 2.7
InnerMappingGroup, 2.7
InterestingLoop, 4.6
Inverse, 2.5
inverse, 2.5
inverse property, 2.10
IsAlternative, 2.10
IsAssociative, 2.10
IsCCLoop, 2.10
IsCLoop, 2.10
IsCommutative, 2.10
IsDiassociative, 2.10
IsDistributive, 2.10
IsEntropic, 2.10
IsExtraLoop, 2.10
IsFlexible, 2.10
IsIdempotent, 2.10
IsLCCLoop, 2.10
IsLCLoop, 2.10
IsLDistributive, 2.10
IsLeftAlternative, 2.10
IsLeftBolLoop, 2.10
IsLeftBruckLoop, 2.10
IsLeftDistributive, 2.10
IsLeftKLoop, 2.10
IsLeftNuclearSquareLoop, 2.10
IsLoop, 2.3
IsLoopCayleyTable, 2.2
IsLoopElement, 2.3
IsLoopTable, 2.2
IsMedial, 2.10
IsMiddleNuclearSquareLoop, 2.10
IsMoufangLoop, 2.10
IsNilpotent, 2.13
IsNormal, 2.11
IsNuclearSquareLoop, 2.10
isomorphism, 1.4
IsomorphismLoops, 3.1
IsomorphismTypeOfMoufangLoop, 4.3
isotopism, 1.4
IsPowerAssociative, 2.10
IsQuasigroup, 2.3
IsQuasigroupCayleyTable, 2.2
IsQuasigroupElement, 2.3
IsQuasigroupTable, 2.2
IsRCCLoop, 2.10
IsRCLoop, 2.10
IsRDistributive, 2.10
IsRightAlternative, 2.10
IsRightBolLoop, 2.10
IsRightBruckLoop, 2.10
IsRightDistributive, 2.10
IsRightKLoop, 2.10
IsRightNuclearSquareLoop, 2.10
IsSemisymmetric, 2.10
IsSimple, 2.11
IsSolvable, 2.14
IsSteinerLoop, 2.10
IsSteinerQuasigroup, 2.10
IsStronglyNilpotent, 2.13
IsSubloop, 2.8
IsSubquasigroup, 2.8
IsTotallySymmetric, 2.10
IsUnipotent, 2.10

K-loop, 2.10
Latin square, 1.4
left Bruck loop, 2.10
left conjugacy closed loop, 2.10
left division, 2.5
left inverse, 2.5
left inverse property, 2.10
left nucleus, 2.9
left section, 1.4
left translation, 1.4
LeftBolLoop, 4.1, 4.2
LeftDivision, 2.5
LeftInverse, 2.5
LeftMultiplicationGroup, 2.7
LeftNucleus, 2.9
LeftSection, 2.7
LeftTranslation, 2.7
LibraryLoop, 4.1
list of files, 1.1
loop, 1.4
LoopByCayleyTable, 2.2
LoopByCyclicModification, 3.2
LoopByDihedralModification, 3.2
LoopFromFile, 2.2
LoopMG2, 3.2
loops of Bol-Moufang type, 2.10

magma, 1.4
middle nucleus, 2.9
MiddleNucleus, 2.9
monoassociative, 2.5
monoid, 1.4
Moufang center, 2.9
Moufang modifications, 3.2
MoufangLoop, 4.3
multiplication group, 1.4
MultiplicationGroup, 2.7
MultiplicativeNeutralElement, 2.4
MyLibraryLoop, 4.1
NaturalHomomorphismByNormalSubloop, 2.12
neutral element, 1.4
NilpotencyClassOfLoop, 2.13
normal closure, 2.11
normal subloop, 2.11
NormalClosure, 2.11
normalized Latin square, 2.2
NormalizedQuasigroupTable, 2.2
Nuc, 2.9
nuclear square loop, 2.10
nucleus, 2.9
NucleusOfLoop, 2.9
NucleusOfQuasigroup, 2.9

octonions, 4.3
One, 2.4
Opposite, 2.2
opposite quasigroup, 2.2
Paige loop, 4.5
PaigeLoop, 4.5
Parent, 2.8
PosInParent, 2.8
power associative, 2.5
power associativity, 2.10
principal loop isotope, 1.4
PrincipalLoopIsotope, 2.2
quasigroup, 1.4
QuasigroupByCayleyTable, 2.2
QuasigroupFromFile, 2.2
RelativeLeftMultiplicationGroup, 2.7
RelativeMultiplicationGroup, 2.7
RelativeRightMultiplicationGroup, 2.7
right Bruck loop, 2.10
right conjugacy closed loop, 2.10
right distributive quasigroup, 2.10
right division, 2.5
right inverse, 2.5
right inverse property, 2.10
right multiplication group, 1.4
right nucleus, 2.9
right section, 1.4
right translation, 1.4
RightDivision, 2.5
RightInverse, 2.5
RightMultiplicationGroup, 2.7
RightNucleus, 2.9
RightSection, 2.7
RightTranslation, 2.7

sedenions, 4.6
semigroup, 1.4
SetLoopElmName, 2.1
SetQuasigroupElmName, 2.1
simple loop, 2.11
Size, 2.4
SmallGeneratingSet, 2.6
Steiner loop, 2.10
Steiner quasigroup, 2.10
SteinerLoop, 4.4
Subloop, 2.8
Subquasigroup, 2.8
TrialityPcGroup, 3.3
TrialityPermGroup, 3.3
two-sided inverse, 1.4
two-sided inverses, 2.10
unipotent quasigroup, 2.10

File translated from TEX by TTH, version 3.61.
On 7 Nov 2004, 17:40.