  
  [1X8 [33X[0;0YSpecific Methods[133X[101X
  
  [33X[0;0YThis  chapter  describes  methods of [5XLOOPS[105X that apply to specific classes of
  loops, mostly Bol and Moufang loops.[133X
  
  
  [1X8.1 [33X[0;0YCore Methods for Bol Loops[133X[101X
  
  
  [1X8.1-1 [33X[0;0YAssociatedLeftBruckLoop and AssociatedRightBruckLoop[133X[101X
  
  [33X[1;0Y[29X[2XAssociatedLeftBruckLoop[102X( [3XQ[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XAssociatedRightBruckLoop[102X( [3XQ[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YThe  left  (resp.  right)  Bruck  loop  associated with a uniquely
            2-divisible left (resp. right) Bol loop [3XQ[103X.[133X
  
  [33X[0;0YLet [22XQ[122X be a left Bol loop such that the mapping [22Xx↦ x^2[122X is a permutation of [22XQ[122X.
  Define  a  new  operation [22X*[122X on [22XQ[122X by [22Xx*y =(x(y^2x))^1/2[122X. Then [22X(Q,*)[122X is a left
  Bruck  loop, called the [13Xassociated left Bruck loop[113X. (In fact, Bruck used the
  isomorphic  operation  [22Xx*y  =  x^1/2(yx^1/2)[122X  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.[133X
  
  [1X8.1-2 IsExactGroupFactorization[101X
  
  [33X[1;0Y[29X[2XIsExactGroupFactorization[102X( [3XG[103X, [3XH1[103X, [3XH2[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X if ([3XG[103X, [3XH1[103X, [3XH2[103X) is an exact group factorization.[133X
  
  [33X[0;0YMany right Bol loops can be constructed from exact group factorizations. The
  triple [22X(G,H_1,H_2)[122X is an [13Xexact group factorization[113X if [22XH_1[122X, [22XH_2[122X are subgroups
  of [22XG[122X such that [22XH_1H_2=G[122X and [22XH_1∩ H_2=1[122X.[133X
  
  [1X8.1-3 RightBolLoopByExactGroupFactorization[101X
  
  [33X[0;0YIf [22X(G,H_1,H_2)[122X is an exact group factorization then [22X(G× G, H_1× H_2, T)[122X with
  [22XT={(x,x^-1)| x∈ G}[122X is a loop folder that gives rise to a right Bol loop.[133X
  
  [33X[1;0Y[29X[2XRightBolLoopByExactGroupFactorization[102X( [3Xarg[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10YThe  right Bol loop constructed from an exact group factorization.
            The  argument  [3Xarg[103X  can  either  be  an  exact group factorization
            [10X[G,H1,H2][110X, or the tuple [10X[G,H][110X, where [10XH[110X is a regular subgroup of [10XG[110X.
            We  also  allow  [3Xarg[103X  to be separate entries rather than a list of
            entries.[133X
  
  
  [1X8.2 [33X[0;0YMoufang Modifications[133X[101X
  
  [33X[0;0YDrápal  [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 [13XMoufang modifications[113X.
  The  library  of  Moufang  loops  included  in  [5XLOOPS[105X  is  based  on Moufang
  modifications. See [DV06] for details.[133X
  
  [1X8.2-1 LoopByCyclicModification[101X
  
  [33X[1;0Y[29X[2XLoopByCyclicModification[102X( [3XQ[103X, [3XS[103X, [3Xa[103X, [3Xh[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10YThe  cyclic  modification of a Moufang loop [3XQ[103X obtained from [3XS[103X, [3Xa[103X[22X=α[122X
            and [3Xh[103X described below.[133X
  
  [33X[0;0YAssume  that  [22XQ[122X is a Moufang loop with a normal subloop [22XS[122X such that [22XQ/S[122X is a
  cyclic  group  of  order [22X2m[122X. Let [22Xh∈ S∩ Z(L)[122X. Let [22Xα[122X be a generator of [22XQ/S[122X and
  write  [22XQ  =  ⋃_i∈  M α^i[122X, where [22XM={-m+1[122X, [22Xdots[122X, [22Xm}[122X. Let [22Xσ:Z-> M[122X be defined by
  [22Xσ(i)=0[122X  if  [22Xi∈  M[122X,  [22Xσ(i)=1[122X  if  [22Xi>m[122X,  and [22Xσ(i)=-1[122X if [22Xi<-m+1[122X. Introduce a new
  multiplication  [22X*[122X on [22XQ[122X by [22Xx*y = xyh^σ(i+j)[122X, where [22Xx∈ α^i[122X, [22Xy∈α^j[122X, [22Xi∈ M[122X and [22Xj∈
  M[122X. Then [22X(Q,*)[122X is a Moufang loop, a [13Xcyclic modification[113X of [22XQ[122X.[133X
  
  [1X8.2-2 LoopByDihedralModification[101X
  
  [33X[1;0Y[29X[2XLoopByDihedralModification[102X( [3XQ[103X, [3XS[103X, [3Xe[103X, [3Xf[103X, [3Xh[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10YThe  dihedral modification of a Moufang loop [3XQ[103X obtained from [3XS[103X, [3Xe[103X,
            [3Xf[103X and [3Xh[103X as described below.[133X
  
  [33X[0;0YLet  [22XQ[122X be a Moufang loop with a normal subloop [22XS[122X such that [22XQ/S[122X is a dihedral
  group  of order [22X4m[122X, with [22Xm≥ 1[122X. Let [22XM[122X and [22Xσ[122X be defined as in the cyclic case.
  Let  [22Xβ[122X,  [22Xγ[122X  be  two  involutions  of  [22XQ/S[122X  such that [22Xα=βγ[122X generates a cyclic
  subgroup  of  [22XQ/S[122X  of  order [22X2m[122X. Let [22Xe∈β[122X and [22Xf∈γ[122X be arbitrary. Then [22XQ[122X can be
  written  as  a  disjoint  union  [22XQ=⋃_i∈ M(α^i∪ eα^i)[122X, and also [22XQ=⋃_i∈ M(α^i∪
  α^if)[122X.  Let [22XG_0=⋃_i∈ Mα^i[122X, and [22XG_1=L∖ G_0[122X. Let [22Xh∈ S∩ N(L)∩ Z(G_0)[122X. Introduce
  a  new  multiplication  [22X*[122X on [22XQ[122X by [22Xx*y = xyh^(-1)^rσ(i+j)[122X, where [22Xx∈α^i∪ eα^i[122X,
  [22Xy∈α^j∪ α^jf[122X, [22Xi∈ M[122X, [22Xj∈ M[122X, [22Xy∈ G_r[122X and [22Xr∈{0,1}[122X. Then [22X(Q,*)[122X is a Moufang loop, a
  [13Xdihedral modification[113X of [22XQ[122X.[133X
  
  [1X8.2-3 LoopMG2[101X
  
  [33X[1;0Y[29X[2XLoopMG2[102X( [3XG[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10YThe Chein loop constructed from a group [3XG[103X.[133X
  
  [33X[0;0YLet  [22XG[122X  be  a  group.  Let  [22XoverlineG={overlineg|g∈ G}[122X be a disjoint copy of
  elements  of  [22XG[122X.  Define  multiplication  [22X*[122X  on  [22XQ=G∪ overlineG[122X by [22Xg*h = gh[122X,
  [22Xg*overlineh=overlinehg[122X,      [22Xoverlineg*h      =      overlinegh^-1}[122X      and
  [22Xoverlineg*overlineh=h^-1g[122X,  where  [22Xg[122X, [22Xh∈ G[122X. Then [22X(Q,*)=M(G,2)[122X is a so-called
  [13XChein  loop[113X,  which  is  always a Moufang loop, and it is associative if and
  only if [22XG[122X is commutative.[133X
  
  
  [1X8.3 [33X[0;0YTriality for Moufang Loops[133X[101X
  
  [33X[0;0YLet  [22XG[122X  be  a group and [22Xσ[122X, [22Xρ[122X be automorphisms of [22XG[122X satisfying [22Xσ^2 = ρ^3 = (σ
  ρ)^2  =  1[122X. Below we write automorphisms as exponents and [22X[g,σ][122X for [22Xg^-1g^σ[122X.
  We  say  that  the triple [22X(G,ρ,σ)[122X is a [13Xgroup with triality[113X if [22X[g, σ] [g,σ]^ρ
  [g,σ]^ρ^2 =1[122X holds for all [22Xg ∈ G[122X. It is known that one can associate a group
  with  triality  [22X(G,ρ,σ)[122X in a canonical way with a Moufang loop [22XQ[122X. See [NV03]
  for more details.[133X
  
  [33X[0;0YFor any Moufang loop [22XQ[122X, we can calculate the triality group as a permutation
  group acting on [22X3|Q|[122X points. If the multiplication group of [22XQ[122X is polycyclic,
  then  we can also represent the triality group as a pc group. In both cases,
  the automorphisms [22Xσ[122X and [22Xρ[122X are in the same family as the elements of [22XG[122X.[133X
  
  [1X8.3-1 TrialityPermGroup[101X
  
  [33X[1;0Y[29X[2XTrialityPermGroup[102X( [3XQ[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10YA  record  with components [10XG[110X, [10Xrho[110X, [10Xsigma[110X, where [10XG[110X is the canonical
            group  with  triality  associated  with a Moufang loop [3XQ[103X, and [10Xrho[110X,
            [10Xsigma[110X are the corresponding triality automorphisms.[133X
  
  [1X8.3-2 TrialityPcGroup[101X
  
  [33X[1;0Y[29X[2XTrialityPcGroup[102X( [3XQ[103X ) [32X function[133X
  
  [33X[0;0YThis  is  a  variation  of  [10XTrialityPermGroup[110X in which [10XG[110X is returned as a pc
  group.[133X
  
  
  [1X8.4 [33X[0;0YRealizing Groups as Multiplication Groups of Loops[133X[101X
  
  [33X[0;0YIt is difficult to determine which groups can occur as multiplication groups
  of loops.[133X
  
  [33X[0;0YThe  following  operations  search for loops whose multiplication groups are
  contained  within  a  specified transitive permutation group [3XG[103X. In all these
  operations,  one can speed up the search by increasing the optional argument
  [3Xdepth[103X,  the price being a much higher memory consumption. The argument [3Xdepth[103X
  is  optimally  chosen  if  in  the  permutation  group  [3XG[103X there are not many
  permutations  fixing  [3Xdepth[103X elements. It is safe to omit the argument or set
  it equal to 2.[133X
  
  [33X[0;0YThe  optional  argument  [3Xinfolevel[103X  determines  the  amount  of  information
  displayed  during  the search. With [10X[3Xinfolevel[103X[10X=0[110X, no information is provided.
  With  [10X[3Xinfolevel[103X[10X=1[110X,  you  get  some  information  on  timing  and  hits. With
  [10X[3Xinfolevel[103X[10X=2[110X, the results are printed as well.[133X
  
  [1X8.4-1 AllLoopTablesInGroup[101X
  
  [33X[1;0Y[29X[2XAllLoopTablesInGroup[102X( [3XG[103X[, [3Xdepth[103X[, [3Xinfolevel[103X]] ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YAll Cayley tables of loops whose multiplication group is contained
            in the transitive permutation group [3XG[103X.[133X
  
  [1X8.4-2 AllProperLoopTablesInGroup[101X
  
  [33X[1;0Y[29X[2XAllProperLoopTablesInGroup[102X( [3XG[103X[, [3Xdepth[103X[, [3Xinfolevel[103X]] ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YAll  Cayley  tables  of  nonassociative loops whose multiplication
            group is contained in the transitive permutation group [3XG[103X.[133X
  
  [1X8.4-3 OneLoopTableInGroup[101X
  
  [33X[1;0Y[29X[2XOneLoopTableInGroup[102X( [3XG[103X[, [3Xdepth[103X[, [3Xinfolevel[103X]] ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA  Cayley  table of a loop whose multiplication group is contained
            in the transitive permutation group [3XG[103X.[133X
  
  [1X8.4-4 OneProperLoopTableInGroup[101X
  
  [33X[1;0Y[29X[2XOneProperLoopTableInGroup[102X( [3XG[103X[, [3Xdepth[103X[, [3Xinfolevel[103X]] ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA Cayley table of a nonassociative loop whose multiplication group
            is contained in the transitive permutation group [3XG[103X.[133X
  
  [1X8.4-5 AllLoopsWithMltGroup[101X
  
  [33X[1;0Y[29X[2XAllLoopsWithMltGroup[102X( [3XG[103X[, [3Xdepth[103X[, [3Xinfolevel[103X]] ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA list of all loops (given as sections) whose multiplication group
            is equal to the transitive permutation group [3XG[103X.[133X
  
  [1X8.4-6 OneLoopWithMltGroup[101X
  
  [33X[1;0Y[29X[2XOneLoopWithMltGroup[102X( [3XG[103X[, [3Xdepth[103X[, [3Xinfolevel[103X]] ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YOne  loop (given as a section) whose multiplication group is equal
            to the transitive permutation group [3XG[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:=PGL(3,3);[127X[104X
    [4X[28XGroup([ (6,7)(8,11)(9,13)(10,12), (1,2,5,7,13,3,8,6,10,9,12,4,11) ])[128X[104X
    [4X[25Xgap>[125X [27Xa:=AllLoopTablesInGroup(g,3,0);; Size(a);[127X[104X
    [4X[28X56[128X[104X
    [4X[25Xgap>[125X [27Xa:=AllLoopsWithMltGroup(g,3,0);; Size(a);[127X[104X
    [4X[28X52[128X[104X
  [4X[32X[104X
  
