loops/doc/chap8.txt

  
  8 Specific Methods
  
  This  chapter  describes  methods of LOOPS that apply to specific classes of
  loops, mostly Bol and Moufang loops.
  
  
  8.1 Core Methods for Bol Loops
  
  
  8.1-1 AssociatedLeftBruckLoop and AssociatedRightBruckLoop
  
  AssociatedLeftBruckLoop( Q )  attribute
  AssociatedRightBruckLoop( Q )  attribute
  Returns:  The  left  (resp.  right)  Bruck  loop  associated with a uniquely
            2-divisible left (resp. right) Bol loop Q.
  
  Let Q be a left Bol loop such that the mapping x↦ x^2 is a permutation of Q.
  Define  a  new  operation * on Q by x*y =(x(y^2x))^1/2. Then (Q,*) is a left
  Bruck  loop, called the associated left Bruck loop. (In fact, Bruck used the
  isomorphic  operation  x*y  =  x^1/2(yx^1/2)  instead.  Our approach is more
  natural  in  the sense that the left Bruck loop associated with a left Bruck
  loop  is  identical  to the original loop.) Associated right Bruck loops are
  defined dually.
  
  8.1-2 IsExactGroupFactorization
  
  IsExactGroupFactorization( G, H1, H2 )  operation
  Returns:  true if (G, H1, H2) is an exact group factorization.
  
  Many right Bol loops can be constructed from exact group factorizations. The
  triple (G,H_1,H_2) is an exact group factorization if H_1, H_2 are subgroups
  of G such that H_1H_2=G and H_1∩ H_2=1.
  
  8.1-3 RightBolLoopByExactGroupFactorization
  
  If (G,H_1,H_2) is an exact group factorization then (G× G, H_1× H_2, T) with
  T={(x,x^-1)| x∈ G} is a loop folder that gives rise to a right Bol loop.
  
  RightBolLoopByExactGroupFactorization( arg )  function
  Returns:  The  right Bol loop constructed from an exact group factorization.
            The  argument  arg  can  either  be  an  exact group factorization
            [G,H1,H2], or the tuple [G,H], where H is a regular subgroup of G.
            We  also  allow  arg  to be separate entries rather than a list of
            entries.
  
  
  8.2 Moufang Modifications
  
  Drápal  [Drá03]  described  two  prominent families of extensions of Moufang
  loops.   It   turns   out  that  these  extensions  suffice  to  obtain  all
  nonassociative  Moufang  loops  of  order  at  most  64  if  one starts with
  so-called  Chein loops. We call the two constructions Moufang modifications.
  The  library  of  Moufang  loops  included  in  LOOPS  is  based  on Moufang
  modifications. See [DV06] for details.
  
  8.2-1 LoopByCyclicModification
  
  LoopByCyclicModification( Q, S, a, h )  function
  Returns:  The  cyclic  modification of a Moufang loop Q obtained from S, a=α
            and h described below.
  
  Assume  that  Q is a Moufang loop with a normal subloop S such that Q/S is a
  cyclic  group  of  order 2m. Let h∈ S∩ Z(L). Let α be a generator of Q/S and
  write  Q  =  ⋃_i∈  M α^i, where M={-m+1, dots, m}. Let σ:Z-> M be defined by
  σ(i)=0  if  i∈  M,  σ(i)=1  if  i>m,  and σ(i)=-1 if i<-m+1. Introduce a new
  multiplication  * on Q by x*y = xyh^σ(i+j), where x∈ α^i, y∈α^j, i∈ M and j∈
  M. Then (Q,*) is a Moufang loop, a cyclic modification of Q.
  
  8.2-2 LoopByDihedralModification
  
  LoopByDihedralModification( Q, S, e, f, h )  function
  Returns:  The  dihedral modification of a Moufang loop Q obtained from S, e,
            f and h as described below.
  
  Let  Q be a Moufang loop with a normal subloop S such that Q/S is a dihedral
  group of order 4m, with mge 1. Let M and σ be defined as in the cyclic case.
  Let  β,  γ  be  two  involutions  of  Q/S  such that α=βγ generates a cyclic
  subgroup  of  Q/S  of  order 2m. Let e∈β and f∈γ be arbitrary. Then Q can be
  written  as  a  disjoint  union  Q=⋃_i∈ M(α^i∪ eα^i), and also Q=⋃_i∈ M(α^i∪
  α^if).  Let G_0=⋃_i∈ Mα^i, and G_1=L∖ G_0. Let h∈ S∩ N(L)∩ Z(G_0). Introduce
  a  new  multiplication  * on Q by x*y = xyh^(-1)^rσ(i+j), where x∈α^i∪ eα^i,
  y∈α^j∪ α^jf, i∈ M, j∈ M, y∈ G_r and r∈{0,1}. Then (Q,*) is a Moufang loop, a
  dihedral modification of Q.
  
  8.2-3 LoopMG2
  
  LoopMG2( G )  function
  Returns:  The Chein loop constructed from a group G.
  
  Let  G  be  a  group.  Let  overlineG={overlineg|g∈ G} be a disjoint copy of
  elements  of  G.  Define  multiplication  *  on  Q=G∪ overlineG by g*h = gh,
  g*overlineh=overlinehg,      overlineg*h      =      overlinegh^-1}      and
  overlineg*overlineh=h^-1g,  where  g, h∈ G. Then (Q,*)=M(G,2) is a so-called
  Chein  loop,  which  is  always a Moufang loop, and it is associative if and
  only if G is commutative.
  
  
  8.3 Triality for Moufang Loops
  
  Let  G  be  a group and σ, ρ be automorphisms of G satisfying σ^2 = ρ^3 = (σ
  ρ)^2  =  1. Below we write automorphisms as exponents and [g,σ] for g^-1g^σ.
  We  say  that  the triple (G,ρ,σ) is a group with triality if [g, σ] [g,σ]^ρ
  [g,σ]^ρ^2 =1 holds for all g ∈ G. It is known that one can associate a group
  with  triality  (G,ρ,σ) in a canonical way with a Moufang loop Q. See [NV03]
  for more details.
  
  For any Moufang loop Q, we can calculate the triality group as a permutation
  group acting on 3|Q| points. If the multiplication group of Q is polycyclic,
  then  we can also represent the triality group as a pc group. In both cases,
  the automorphisms σ and ρ are in the same family as the elements of G.
  
  8.3-1 TrialityPermGroup
  
  TrialityPermGroup( Q )  function
  Returns:  A  record  with components G, rho, sigma, where G is the canonical
            group  with  triality  associated  with a Moufang loop Q, and rho,
            sigma are the corresponding triality automorphisms.
  
  8.3-2 TrialityPcGroup
  
  TrialityPcGroup( Q )  function
  
  This  is  a  variation  of  TrialityPermGroup in which G is returned as a pc
  group.
  
  
  8.4 Realizing Groups as Multiplication Groups of Loops
  
  It is difficult to determine which groups can occur as multiplication groups
  of loops.
  
  The  following  operations  search for loops whose multiplication groups are
  contained  within  a  specified transitive permutation group G. In all these
  operations,  one can speed up the search by increasing the optional argument
  depth,  the price being a much higher memory consumption. The argument depth
  is  optimally  chosen  if  in  the  permutation  group  G there are not many
  permutations  fixing  depth elements. It is safe to omit the argument or set
  it equal to 2.
  
  The  optional  argument  infolevel  determines  the  amount  of  information
  displayed  during  the search. With infolevel=0, no information is provided.
  With  infolevel=1,  you  get  some  information  on  timing  and  hits. With
  infolevel=2, the results are printed as well.
  
  8.4-1 AllLoopTablesInGroup
  
  AllLoopTablesInGroup( G[, depth[, infolevel]] )  operation
  Returns:  All Cayley tables of loops whose multiplication group is contained
            in the transitive permutation group G.
  
  8.4-2 AllProperLoopTablesInGroup
  
  AllProperLoopTablesInGroup( G[, depth[, infolevel]] )  operation
  Returns:  All  Cayley  tables  of  nonassociative loops whose multiplication
            group is contained in the transitive permutation group G.
  
  8.4-3 OneLoopTableInGroup
  
  OneLoopTableInGroup( G[, depth[, infolevel]] )  operation
  Returns:  A  Cayley  table of a loop whose multiplication group is contained
            in the transitive permutation group G.
  
  8.4-4 OneProperLoopTableInGroup
  
  OneProperLoopTableInGroup( G[, depth[, infolevel]] )  operation
  Returns:  A Cayley table of a nonassociative loop whose multiplication group
            is contained in the transitive permutation group G.
  
  8.4-5 AllLoopsWithMltGroup
  
  AllLoopsWithMltGroup( G[, depth[, infolevel]] )  operation
  Returns:  A list of all loops (given as sections) whose multiplication group
            is equal to the transitive permutation group G.
  
  8.4-6 OneLoopWithMltGroup
  
  OneLoopWithMltGroup( G[, depth[, infolevel]] )  operation
  Returns:  One  loop (given as a section) whose multiplication group is equal
            to the transitive permutation group G.
  
    Example  
    gap> g:=PGL(3,3);

    Group([ (6,7)(8,11)(9,13)(10,12), (1,2,5,7,13,3,8,6,10,9,12,4,11) ])

    gap> a:=AllLoopTablesInGroup(g,3,0);; Size(a);

    56

    gap> a:=AllLoopsWithMltGroup(g,3,0);; Size(a);

    52