  
  [1X3 [33X[0;0YStrong Shoda pairs[133X[101X
  
  
  [1X3.1 [33X[0;0YComputing strong Shoda pairs[133X[101X
  
  [1X3.1-1 StrongShodaPairs[101X
  
  [29X[2XStrongShodaPairs[102X( [3XG[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10YA list of pairs of subgroups of the input group.[133X
  
  [33X[0;0YThe input should be a finite group [3XG[103X.[133X
  
  [33X[0;0YComputes  a  list  of  representatives  of the equivalence classes of [13Xstrong
  Shoda pairs[113X ([14X9.15[114X) of a finite group [3XG[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XStrongShodaPairs( SymmetricGroup(4) );[127X[104X
    [4X[28X[ [ Sym( [ 1 .. 4 ] ), Group([ (1,4)(2,3), (1,3)(2,4), (2,4,3), (3,4) ]) ],[128X[104X
    [4X[28X  [ Sym( [ 1 .. 4 ] ), Group([ (1,4)(2,3), (1,3)(2,4), (2,4,3) ]) ],[128X[104X
    [4X[28X  [ Group([ (3,4), (1,3,2,4) ]), Group([ (1,3,2,4), (1,2)(3,4) ]) ],[128X[104X
    [4X[28X  [ Group([ (1,3,2,4), (3,4) ]), Group([ (3,4), (1,2)(3,4) ]) ],[128X[104X
    [4X[28X  [ Group([ (2,4,3), (1,4)(2,3) ]), Group([ (1,4)(2,3), (1,3)(2,4) ]) ] ][128X[104X
    [4X[25Xgap>[125X [27XStrongShodaPairs( DihedralGroup(64) );[127X[104X
    [4X[28X[ [ <pc group of size 64 with 6 generators>,[128X[104X
    [4X[28X      Group([ f6, f5, f4, f3, f1, f2 ]) ],[128X[104X
    [4X[28X  [ <pc group of size 64 with 6 generators>, Group([ f6, f5, f4, f3, f1*f2 ])[128X[104X
    [4X[28X     ],[128X[104X
    [4X[28X  [ <pc group of size 64 with 6 generators>, Group([ f6, f5, f4, f3, f2 ]) ],[128X[104X
    [4X[28X  [ <pc group of size 64 with 6 generators>, Group([ f6, f5, f4, f3, f1 ]) ],[128X[104X
    [4X[28X  [ Group([ f1*f2, f4*f5*f6, f5*f6, f6, f3, f3 ]),[128X[104X
    [4X[28X      Group([ f6, f5, f4, f1*f2 ]) ],[128X[104X
    [4X[28X  [ Group([ f6, f5, f2, f3, f4 ]), Group([ f6, f5 ]) ],[128X[104X
    [4X[28X  [ Group([ f6, f2, f3, f4, f5 ]), Group([ f6 ]) ],[128X[104X
    [4X[28X  [ Group([ f2, f3, f4, f5, f6 ]), Group([  ]) ] ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X3.2 [33X[0;0YProperties related with Shoda pairs[133X[101X
  
  [1X3.2-1 IsStrongShodaPair[101X
  
  [29X[2XIsStrongShodaPair[102X( [3XG[103X, [3XK[103X, [3XH[103X ) [32X operation
  
  [33X[0;0YThe first argument should be a finite group [3XG[103X, the second one a sugroup [3XK[103X of
  [3XG[103X and the third one a subgroup of [3XK[103X.[133X
  
  [33X[0;0YReturns  [9Xtrue[109X  if  ([3XK[103X,[3XH[103X)  is  a  [13Xstrong  Shoda  pair[113X  ([14X9.15[114X) of [3XG[103X, and [9Xfalse[109X
  otherwise.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XG:=SymmetricGroup(3);; K:=Group([(1,2,3)]);; H:=Group( () );;[127X[104X
    [4X[25Xgap>[125X [27XIsStrongShodaPair( G, K, H );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsStrongShodaPair( G, G, H );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsStrongShodaPair( G, K, K );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsStrongShodaPair( G, G, K );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X3.2-2 IsShodaPair[101X
  
  [29X[2XIsShodaPair[102X( [3XG[103X, [3XK[103X, [3XH[103X ) [32X operation
  
  [33X[0;0YThe  first argument should be a finite group [3XG[103X, the second a subgroup [3XK[103X of [3XG[103X
  and the third one a subgroup of [3XK[103X.[133X
  
  [33X[0;0YReturns [9Xtrue[109X if ([3XK[103X,[3XH[103X) is a [13XShoda pair[113X ([14X9.14[114X) of [3XG[103X.[133X
  
  [33X[0;0YNote  that  every strong Shoda pair is a Shoda pair, but the converse is not
  true.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XG:=AlternatingGroup(5);;[127X[104X
    [4X[25Xgap>[125X [27XK:=AlternatingGroup(4);;[127X[104X
    [4X[25Xgap>[125X [27XH := Group( (1,2)(3,4), (1,3)(2,4) );;[127X[104X
    [4X[25Xgap>[125X [27XIsStrongShodaPair( G, K, H );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsShodaPair( G, K, H );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X3.2-3 IsStronglyMonomial[101X
  
  [29X[2XIsStronglyMonomial[102X( [3XG[103X ) [32X operation
  
  [33X[0;0YThe input [3XG[103X should be a finite group.[133X
  
  [33X[0;0YReturns [9Xtrue[109X if [3XG[103X is a [13Xstrongly monomial[113X ([14X9.16[114X) finite group.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XS4:=SymmetricGroup(4);;[127X[104X
    [4X[25Xgap>[125X [27XIsStronglyMonomial(S4);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XG:=SmallGroup(24,3);;[127X[104X
    [4X[25Xgap>[125X [27XIsStronglyMonomial(G);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsMonomial(G);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XG:=SmallGroup(1000,86);;[127X[104X
    [4X[25Xgap>[125X [27XIsMonomial(G);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsStronglyMonomial(G);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
