loops/doc/chap9.txt

  
  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
        sth 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 Left Bruck Loops and Right Bruck Loops
  
  The  emmerging  library  named  left  Bruck contains all left Bruck loops of
  orders 3, 9, 27 and 81 (there are 1, 2, 7 and 72 such loops, respectively).
  
  For  an  odd  prime p, left Bruck loops of order p^k are centrally nilpotent
  and  hence central extensions of the cyclic group of order p by a left Bruck
  loop  of  order  p^k-1. It is known that left Bruck loops of order p and p^2
  are  abelian  groups;  we  have  included them in the library because of the
  iterative nature of the construction of nilpotent loops.
  
  9.3-1 LeftBruckLoop
  
  LeftBruckLoop( n, m )  function
  Returns:  The mth left Bruck loop of order n in the library.
  
  9.3-2 RightBruckLoop
  
  RightBruckLoop( n, m )  function
  Returns:  The mth right Bruck loop of order n in the library.
  
  
  9.4 Moufang Loops
  
  The library named Moufang contains all nonassociative Moufang loops of order
  nle 64 and n∈{81,243}.
  
  9.4-1 MoufangLoop
  
  MoufangLoop( n, m )  function
  Returns:  The mth Moufang loop of order n in the library.
  
  For  nle  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).
  
  
  9.5 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.
  
  9.5-1 CodeLoop
  
  CodeLoop( n, m )  function
  Returns:  The mth code loop of order n in the library.
  
  
  9.6 Steiner Loops
  
  Here is how the libary named Steiner is described within LOOPS:
  
    Example  
    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].
  
  9.6-1 SteinerLoop
  
  SteinerLoop( n, m )  function
  Returns:  The mth Steiner loop of order n in the library.
  
  
  9.7 Conjugacy Closed Loops
  
  The  library  named  RCC  contains  all nonassocitive right conjugacy closed
  loops  of  order  nle  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 nle 31.
  
  Let Q be a right conjugacy closed loop, G its right multiplication group and
  T  its  right section. Then ⟨ T⟩ = 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} 
  
  
  
  9.7-1 RCCLoop and RightConjugacyClosedLoop
  
  RCCLoop( n, m )  function
  RightConjugacyClosedLoop( n, m )  function
  Returns:  The mth right conjugacy closed loop of order n in the library.
  
  
  9.7-2 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 CC loops of order 2le 2^kle 64, 3le 3^kle
  81,  5le 5^kle 125, 7le 7^kle 343, all nonassociative CC loops of order less
  than  28,  and  all  nonassociative CC loops of order p^2 and 2p for any odd
  prime 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 Z_p^2 and Z_p × 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 Z_p^2 by x⋅ 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
        Z_p^2 by x⋅ y = x + y + kpx^2y.
  
      Case  m  = 3: Define multiplication on Z_p × 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  Z_2 × 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 ).
  
  
  9.7-3 CCLoop and ConjugacyClosedLoop
  
  CCLoop( n, m )  function
  ConjugacyClosedLoop( n, m )  function
  Returns:  The mth conjugacy closed loop of order n in the library.
  
  
  9.8 Small Loops
  
  The  library named small contains all nonassociative loops of order 5 and 6.
  There are 5 and 107 such loops, respectively.
  
  9.8-1 SmallLoop
  
  SmallLoop( n, m )  function
  Returns:  The mth loop of order n in the library.
  
  
  9.9 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.
  
  9.9-1 PaigeLoop
  
  PaigeLoop( q )  function
  Returns:  The  Paige loop constructed over the finite field of order q. Only
            the case q=2 is implemented.
  
  
  9.10 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.
  
  9.10-1 NilpotentLoop
  
  NilpotentLoop( n, m )  function
  Returns:  The mth nilpotent loop of order n in the library.
  
  
  9.11 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)
  and all commutative automorphic loops of order 3, 9, 27 and 81 (there are 1,
  2, 7 and 72 such loops).
  
  It  turns  out  that  commutative automorphic loops of order 3, 9, 27 and 81
  (but  not  243) are in one-to-on correspondence with left Bruck loops of the
  respective orders, see [Gre14], [SV17]. Only the left Bruck loops are stored
  in the library.
  
  9.11-1 AutomorphicLoop
  
  AutomorphicLoop( n, m )  function
  Returns:  The mth automorphic loop of order n in the library.
  
  
  9.12 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.
  
  9.12-1 InterestingLoop
  
  InterestingLoop( n, m )  function
  Returns:  The mth interesting loop of order n in the library.
  
  
  9.13 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).
  
  9.13-1 ItpSmallLoop
  
  ItpSmallLoop( n, m )  function
  Returns:  The mth small loop of order n up to isotopism in the library.
  
    Example  
    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.