  
  [1X3 Strong Shoda pairs[0X
  
  
  [1X3.1 Computing strong Shoda pairs[0X
  
  [1X3.1-1 StrongShodaPairs[0m
  
  [2X> StrongShodaPairs( [0X[3XG[0X[2X ) ___________________________________________[0Xattribute
  [6XReturns:[0X  A list of pairs of subgroups of the input group.
  
  The input should be a finite group [3XG[0m.
  
  Computes  a  list  of  representatives  of the equivalence classes of [13Xstrong
  Shoda pairs[0m ([14X7.15[0m) of a finite group [3XG[0m.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4X[0X
    [4Xgap> StrongShodaPairs( SymmetricGroup(4) );[0X
    [4X[ [ Sym( [ 1 .. 4 ] ), Group([ (1,3)(2,4), (1,4)(2,3), (2,4,3), (1,2) ]) ],[0X
    [4X  [ Sym( [ 1 .. 4 ] ), Group([ (1,3)(2,4), (1,4)(2,3), (2,4,3) ]) ],[0X
    [4X  [ Group([ (1,2)(3,4), (1,3,2,4), (3,4) ]), Group([ (1,2)(3,4), (1,3,2,4) ])[0X
    [4X     ],[0X
    [4X  [ Group([ (1,2)(3,4), (3,4), (1,3,2,4) ]), Group([ (1,2)(3,4), (3,4) ]) ],[0X
    [4X  [ Group([ (1,4)(2,3), (1,3)(2,4), (2,4,3) ]),[0X
    [4X      Group([ (1,4)(2,3), (1,3)(2,4) ]) ] ][0X
    [4Xgap> StrongShodaPairs( DihedralGroup(64) );[0X
    [4X[ [ <pc group of size 64 with 6 generators>,[0X
    [4X      Group([ f6, f5, f4, f3, f1, f2 ]) ],[0X
    [4X  [ <pc group of size 64 with 6 generators>, Group([ f6, f5, f4, f3, f1*f2 ])[0X
    [4X     ],[0X
    [4X  [ <pc group of size 64 with 6 generators>, Group([ f6, f5, f4, f3, f2 ]) ],[0X
    [4X  [ <pc group of size 64 with 6 generators>, Group([ f6, f5, f4, f3, f1 ]) ],[0X
    [4X  [ Group([ f1*f2, f4*f5*f6, f5*f6, f6, f3, f3 ]),[0X
    [4X      Group([ f6, f5, f4, f1*f2 ]) ],[0X
    [4X  [ Group([ f6, f5, f2, f3, f4 ]), Group([ f6, f5 ]) ],[0X
    [4X  [ Group([ f6, f2, f3, f4, f5 ]), Group([ f6 ]) ],[0X
    [4X  [ Group([ f2, f3, f4, f5, f6 ]), Group([  ]) ] ][0X
    [4X[0X
  [4X------------------------------------------------------------------[0X
  
  
  [1X3.2 Properties related with Shoda pairs[0X
  
  [1X3.2-1 IsStrongShodaPair[0m
  
  [2X> IsStrongShodaPair( [0X[3XG, K, H[0X[2X ) ____________________________________[0Xoperation
  
  The first argument should be a finite group [3XG[0m, the second one a sugroup [3XK[0m of
  [3XG[0m and the third one a subgroup of [3XK[0m.
  
  Returns  [9Xtrue[0m  if  ([3XK[0m,[3XH[0m)  is  a  [13Xstrong  Shoda  pair[0m  ([14X7.15[0m) of [3XG[0m, and [9Xfalse[0m
  otherwise.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4X[0X
    [4Xgap> G:=SymmetricGroup(3);; K:=Group([(1,2,3)]);; H:=Group( () );;[0X
    [4Xgap> IsStrongShodaPair( G, K, H );[0X
    [4Xtrue[0X
    [4Xgap> IsStrongShodaPair( G, G, H );[0X
    [4Xfalse[0X
    [4Xgap> IsStrongShodaPair( G, K, K );[0X
    [4Xfalse[0X
    [4Xgap> IsStrongShodaPair( G, G, K );[0X
    [4Xtrue[0X
    [4X[0X
  [4X------------------------------------------------------------------[0X
  
  [1X3.2-2 IsShodaPair[0m
  
  [2X> IsShodaPair( [0X[3XG, K, H[0X[2X ) __________________________________________[0Xoperation
  
  The  first argument should be a finite group [3XG[0m, the second a subgroup [3XK[0m of [3XG[0m
  and the third one a subgroup of [3XK[0m.
  
  Returns [9Xtrue[0m if ([3XK[0m,[3XH[0m) is a [13XShoda pair[0m ([14X7.14[0m) of [3XG[0m.
  
  Note  that  every strong Shoda pair is a Shoda pair, but the converse is not
  true.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4X[0X
    [4Xgap> G:=AlternatingGroup(5);;[0X
    [4Xgap> K:=AlternatingGroup(4);;[0X
    [4Xgap> H := Group( (1,2)(3,4), (1,3)(2,4) );;[0X
    [4Xgap> IsStrongShodaPair( G, K, H );[0X
    [4Xfalse[0X
    [4Xgap> IsShodaPair( G, K, H );[0X
    [4Xtrue[0X
    [4X[0X
  [4X------------------------------------------------------------------[0X
  
  [1X3.2-3 IsStronglyMonomial[0m
  
  [2X> IsStronglyMonomial( [0X[3XG[0X[2X ) _________________________________________[0Xoperation
  
  The input [3XG[0m should be a finite group.
  
  Returns [9Xtrue[0m if [3XG[0m is a [13Xstrongly monomial[0m ([14X7.16[0m) finite group.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4X[0X
    [4Xgap> S4:=SymmetricGroup(4);;[0X
    [4Xgap> IsStronglyMonomial(S4);[0X
    [4Xtrue[0X
    [4Xgap> G:=SmallGroup(24,3);;[0X
    [4Xgap> IsStronglyMonomial(G);[0X
    [4Xfalse[0X
    [4Xgap> IsMonomial(G);[0X
    [4Xfalse[0X
    [4Xgap> G:=SmallGroup(1000,86);;[0X
    [4Xgap> IsMonomial(G);[0X
    [4Xtrue[0X
    [4Xgap> IsStronglyMonomial(G);[0X
    [4Xfalse[0X
    [4X[0X
  [4X------------------------------------------------------------------[0X
  
