GAP (Loops) - Chapter 9: Libraries of Loops

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9 Libraries of Loops
9.1 A Typical Library
9.1-1 LibraryLoop
9.1-2 MyLibraryLoop
9.1-3 DisplayLibraryInfo
9.2 Left Bol Loops and Right Bol Loops
9.2-1 LeftBolLoop
9.2-2 RightBolLoop
9.3 Moufang Loops
9.3-1 MoufangLoop
9.4 Code Loops
9.4-1 CodeLoop
9.5 Steiner Loops
9.5-1 SteinerLoop
9.6 Conjugacy Closed Loops
9.6-1 RCCLoop and RightConjugacyClosedLoop
9.6-2 LCCLoop and LeftConjugacyClosedLoop
9.6-3 CCLoop and ConjugacyClosedLoop
9.7 Small Loops
9.7-1 SmallLoop
9.8 Paige Loops
9.8-1 PaigeLoop
9.9 Nilpotent Loops
9.9-1 NilpotentLoop
9.10 Automorphic Loops
9.10-1 AutomorphicLoop
9.11 Interesting Loops
9.11-1 InterestingLoop
9.12 Libraries of Loops Up To Isotopism
9.12-1 ItpSmallLoop

### 9 Libraries of Loops

Libraries of small loops form an integral part of LOOPS. The loops are stored in libraries up to isomorphism and, sometimes, up to isotopism.

#### 9.1 A Typical Library

A library named my Library is stored in file `data/mylibrary.tbl`, and the corresponding data structure is named `LOOPS_my_library_data`. For example, when the library is called left Bol, the corresponding data file is called `data/leftbol.tbl` and the corresponding data structure is named `LOOPS_left_bol_data`.

In most cases, the array `LOOPS_my_library_data` consists of three lists:

- `LOOPS_my_library_data[1]` is a list of orders for which there is at least one loop in the library,

- `LOOPS_my_library_data[2][k]` is the number of loops of order `LOOPS_my_library_data[1][k]` in the library,

- `LOOPS_my_library_data[3][k][s]` contains data necessary to produce the `s`th loop of order `LOOPS_my_library_data[1][k]` in the library.

The format of `LOOPS_my_library_data[3]` depends heavily on the particular library and is not standardized in any way. The data is often coded to save space.

##### 9.1-1 LibraryLoop

`‣ LibraryLoop`( libname, n, m ) | ( function )

Returns: The mth loop of order n from the library named libname.

##### 9.1-2 MyLibraryLoop

`‣ MyLibraryLoop`( n, m ) | ( function )

This is a template function that retrieves the mth loop of order n from the library named my library.

For example, the mth left Bol loop of order n is obtained via `LeftBolLoop(n,m)` or via `LibraryLoop("left Bol",n,m)`.

##### 9.1-3 DisplayLibraryInfo

`‣ DisplayLibraryInfo`( libname ) | ( function )

Returns: Brief information about the loops contained in the library named libname.

We are now going to describe the individual libraries.

#### 9.2 Left Bol Loops and Right Bol Loops

The library named left Bol contains all nonassociative left Bol loops of order less than 17, including Moufang loops, as well as all left Bol loops of order \(pq\) for primes \(p>q>2\). There are 6 such loops of order 8, 1 of order 12, 2 of order 15, 2038 of order 16, and \((p+q-4)/2\) of order \(pq\).

The classification of left Bol loops of order 16 was first accomplished by Moorhouse [Moo]. Our library was generated independently and it agrees with Moorhouse's results. The left Bol loops of order \(pq\) were classified in [KNV15].

##### 9.2-1 LeftBolLoop

`‣ LeftBolLoop`( n, m ) | ( function )

Returns: The mth left Bol loop of order n in the library.

##### 9.2-2 RightBolLoop

`‣ RightBolLoop`( n, m ) | ( function )

Returns: The mth right Bol loop of order n in the library.

Remark: Only left Bol loops are stored in the library. Right Bol loops are retrieved by calling `Opposite` on left Bol loops.

#### 9.3 Moufang Loops

The library named Moufang contains all nonassociative Moufang loops of order \(n\le 64\) and \(n\in\{81,243\}\).

##### 9.3-1 MoufangLoop

`‣ MoufangLoop`( n, m ) | ( function )

Returns: The mth Moufang loop of order n in the library.

For \(n\le 63\), our catalog numbers coincide with those of Goodaire et al. [GMR99]. The classification of Moufang loops of order 64 and 81 was carried out in [NV07]. The classification of Moufang loops of order 243 was carried out by Slattery and Zenisek [SZ12].

The extent of the library is summarized below:

\[ \begin{array}{r|rrrrrrrrrrrrrrrrrr} order&12&16&20&24&28&32&36&40&42&44&48&52&54&56&60&64&81&243\cr loops&1 &5 &1 &5 &1 &71&4 &5 &1 &1 &51&1 &2 &4 &5 &4262& 5 &72 \end{array} \]

The octonion loop of order 16 (i.e., the multiplication loop of the basis elements in the 8-dimensional standard real octonion algebra) can be obtained as `MoufangLoop(16,3)`.

#### 9.4 Code Loops

The library named code contains all nonassociative code loops of order less than 65. There are 5 such loops of order 16, 16 of order 32, and 80 of order 64, all Moufang. The library merely points to the corresponding Moufang loops. See [NV07] for a classification of small code loops.

##### 9.4-1 CodeLoop

`‣ CodeLoop`( n, m ) | ( function )

Returns: The mth code loop of order n in the library.

#### 9.5 Steiner Loops

Here is how the libary named 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

Our labeling of Steiner loops of order 16 coincides with the labeling of Steiner triple systems of order 15 in [CR99].

##### 9.5-1 SteinerLoop

`‣ SteinerLoop`( n, m ) | ( function )

Returns: The mth Steiner loop of order n in the library.

#### 9.6 Conjugacy Closed Loops

The library named RCC contains all nonassocitive right conjugacy closed loops of order \(n\le 27\) up to isomorphism. The data for the library was generated by Katharina Artic [Art15] who can also provide additional data for all right conjugacy closed loops of order \(n\le 31\).

Let \(Q\) be a right conjugacy closed loop, \(G\) its right multiplication group and \(T\) its right section. Then \(\langle T\rangle = G\) is a transitive group, and \(T\) is a union of conjugacy classes of \(G\). Every right conjugacy closed loop of order \(n\) can therefore be represented as a union of certain conjugacy classes of a transitive group of degree \(n\). This is how right conjugacy closed loops of order less than \(28\) are represented in LOOPS. The following table summarizes the number of right conjugacy closed loops of a given order up to isomorphism:

\[ \begin{array}{r|rrrrrrrrrrrrrrrr} order &6& 8&9&10& 12&14&15& 16& 18& 20&\cr loops &3&19&5&16&155&97& 17&6317&1901&8248&\cr \hline order &21& 22& 24& 25& 26& 27\cr loops &119&10487&471995& 119&151971&152701 \end{array} \]

##### 9.6-1 RCCLoop and RightConjugacyClosedLoop

`‣ RCCLoop`( n, m ) | ( function )

`‣ RightConjugacyClosedLoop`( n, m ) | ( function )

Returns: The mth right conjugacy closed loop of order n in the library.

##### 9.6-2 LCCLoop and LeftConjugacyClosedLoop

`‣ LCCLoop`( n, m ) | ( function )

`‣ LeftConjugacyClosedLoop`( n, m ) | ( function )

Returns: The mth left conjugacy closed loop of order n in the library.

Remark: Only the right conjugacy closed loops are stored in the library. Left conjugacy closed loops are obtained from right conjugacy closed loops via `Opposite`.

The library named CC contains all nonassociative conjugacy closed loops of order \(n\le 27\) and also of orders \(2p\) and \(p^2\) for all primes \(p\).

By results of Kunen [Kun00], for every odd prime \(p\) there are precisely 3 nonassociative conjugacy closed loops of order \(p^2\). Csörgő and Drápal [CD05] described these 3 loops by multiplicative formulas on \(\mathbb{Z}_{p^2}\) and \(\mathbb{Z}_p \times \mathbb{Z}_p\) as follows:

- Case \(m = 1\):Let \(k\) be the smallest positive integer relatively prime to \(p\) and such that \(k\) is a square modulo \(p\) (i.e., \(k=1\)). Define multiplication on \(\mathbb{Z}_{p^2}\) by \(x\cdot y = x + y + kpx^2y\).

- Case \(m = 2\): Let \(k\) be the smallest positive integer relatively prime to \(p\) and such that \(k\) is not a square modulo \(p\). Define multiplication on \(\mathbb{Z}_{p^2}\) by \(x\cdot y = x + y + kpx^2y\).

- Case \(m = 3\): Define multiplication on \(\mathbb{Z}_p \times \mathbb{Z}_p\) by \((x,a)(y,b) = (x+y, a+b+x^2y )\).

Moreover, Wilson [WJ75] constructed a nonassociative conjugacy closed loop of order \(2p\) for every odd prime \(p\), and Kunen [Kun00] showed that there are no other nonassociative conjugacy closed oops of this order. Here is the relevant multiplication formula on \(\mathbb{Z}_2 \times \mathbb{Z}_p\): \((0,m)(0,n) = ( 0, m + n )\), \((0,m)(1,n) = ( 1, -m + n )\), \((1,m)(0,n) = ( 1, m + n)\), \((1,m)(1,n) = ( 0, 1 - m + n )\).

##### 9.6-3 CCLoop and ConjugacyClosedLoop

`‣ CCLoop`( n, m ) | ( function )

`‣ ConjugacyClosedLoop`( n, m ) | ( function )

Returns: The mth conjugacy closed loop of order n in the library.

#### 9.7 Small Loops

The library named small contains all nonassociative loops of order 5 and 6. There are 5 and 107 such loops, respectively.

##### 9.7-1 SmallLoop

`‣ SmallLoop`( n, m ) | ( function )

Returns: The mth loop of order n in the library.

#### 9.8 Paige Loops

Paige loops are nonassociative finite simple Moufang loops. By [Lie87], there is precisely one Paige loop for every finite field.

The library named Paige contains the smallest nonassociative simple Moufang loop.

##### 9.8-1 PaigeLoop

`‣ PaigeLoop`( q ) | ( function )

Returns: The Paige loop constructed over the finite field of order q. Only the case `q=2` is implemented.

#### 9.9 Nilpotent Loops

The library named nilpotent contains all nonassociative nilpotent loops of order less than 12 up to isomorphism. There are 2 nonassociative nilpotent loops of order 6, 134 of order 8, 8 of order 9 and 1043 of order 10.

See [DV09] for more on enumeration of nilpotent loops. For instance, there are 2623755 nilpotent loops of order 12, and 123794003928541545927226368 nilpotent loops of order 22.

##### 9.9-1 NilpotentLoop

`‣ NilpotentLoop`( n, m ) | ( function )

Returns: The mth nilpotent loop of order n in the library.

#### 9.10 Automorphic Loops

The library named automorphic contains all nonassociative automorphic loops of order less than 16 up to isomorphism (there is 1 such loop of order 6, 7 of order 8, 3 of order 10, 2 of order 12, 5 of order 14, and 2 of order 15), all commutative automorphic loops of order 3, 9, 27 and 81 (there are 1, 2, 7 and 72 such loops, respectively, including abelian groups), and commutative automorphic loops \(Q\) of order 243 possessing a central subloop \(S\) of order 3 such that \(Q/S\) is not the elementary abelian group of order 81 (there are 118451 such loops).

##### 9.10-1 AutomorphicLoop

`‣ AutomorphicLoop`( n, m ) | ( function )

Returns: The mth automorphic loop of order n in the library.

#### 9.11 Interesting Loops

The library named interesting contains some loops that are illustrative in the theory of loops. At this point, the library contains a nonassociative loop of order 5, a nonassociative nilpotent loop of order 6, a non-Moufang left Bol loop of order 16, the loop of sedenions of order 32 (sedenions generalize octonions), and the unique nonassociative simple right Bol loop of order 96 and exponent 2.

##### 9.11-1 InterestingLoop

`‣ InterestingLoop`( n, m ) | ( function )

Returns: The mth interesting loop of order n in the library.

#### 9.12 Libraries of Loops Up To Isotopism

For the library named small we also provide the corresponding library of loops up to isotopism. In general, given a library named libname, the corresponding library of loops up to isotopism is named itp lib, and the loops can be retrieved by the template `ItpLibLoop(n,m)`.

##### 9.12-1 ItpSmallLoop

`‣ ItpSmallLoop`( n, m ) | ( function )

Returns: The mth small loop of order n up to isotopism in the library.

    gap> SmallLoop( 6, 14 );

    <small loop 6/14>
    gap> ItpSmallLoop( 6, 14 );

    <small loop 6/42>
    gap> LibraryLoop( "itp small", 6, 14 );

    <small loop 6/42>

Note that loops up to isotopism form a subset of the corresponding library of loops up to isomorphism. For instance, the above example shows that the 14th small loop of order 6 up to isotopism is in fact the 42nd small loop of order 6 up to isomorphism.

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