  
  [1X3 [33X[0;0YShoda pairs[133X[101X
  
  
  [1X3.1 [33X[0;0YComputing extremely strong Shoda pairs[133X[101X
  
  [1X3.1-1 ExtremelyStrongShodaPairs[101X
  
  [33X[1;0Y[29X[2XExtremelyStrongShodaPairs[102X( [3XG[103X ) [32X attribute[133X
  [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 [13Xextremely
  strong Shoda pairs[113X ([14X9.16[114X) of a finite group [3XG[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XExtremelyStrongShodaPairs(DihedralGroup(32));[127X[104X
    [4X[28X[ [ <pc group of size 32 with 5 generators>, [128X[104X
    [4X[28X      <pc group of size 32 with 5 generators> ], [128X[104X
    [4X[28X  [ <pc group of size 32 with 5 generators>, [128X[104X
    [4X[28X      Group([ f1*f2*f3*f4*f5, f3, f4, f5 ]) ], [128X[104X
    [4X[28X  [ <pc group of size 32 with 5 generators>, Group([ f2, f3, f4, f5 ]) ], [128X[104X
    [4X[28X  [ <pc group of size 32 with 5 generators>, Group([ f1, f3, f4, f5 ]) ], [128X[104X
    [4X[28X  [ Group([ f1*f2*f3*f4*f5, f3, f4, f5 ]), Group([ f1*f2*f4*f5, f4, f5 ]) ], [128X[104X
    [4X[28X  [ Group([ f2, f3, f4, f5 ]), Group([ f5 ]) ], [128X[104X
    [4X[28X  [ Group([ f2, f3, f4, f5 ]), Group([  ]) ] ][128X[104X
    [4X[25Xgap>[125X [27XExtremelyStrongShodaPairs(SL(2,3));          [127X[104X
    [4X[28X[ [ SL(2,3), SL(2,3) ], [128X[104X
    [4X[28X  [ SL(2,3), [128X[104X
    [4X[28X      Group([ [ [ Z(3)^0, Z(3)^0 ], [ Z(3)^0, Z(3) ] ], [128X[104X
    [4X[28X          [ [ Z(3), Z(3)^0 ], [ Z(3)^0, Z(3)^0 ] ], [128X[104X
    [4X[28X          [ [ Z(3), 0*Z(3) ], [ 0*Z(3), Z(3) ] ] ]) ], [128X[104X
    [4X[28X  [ [128X[104X
    [4X[28X      Group([ [ [ Z(3)^0, Z(3)^0 ], [ Z(3)^0, Z(3) ] ], [128X[104X
    [4X[28X          [ [ Z(3), Z(3)^0 ], [ Z(3)^0, Z(3)^0 ] ], [128X[104X
    [4X[28X          [ [ Z(3), 0*Z(3) ], [ 0*Z(3), Z(3) ] ] ]), [128X[104X
    [4X[28X      Group([ [ [ 0*Z(3), Z(3) ], [ Z(3)^0, 0*Z(3) ] ], [128X[104X
    [4X[28X          [ [ Z(3), 0*Z(3) ], [ 0*Z(3), Z(3) ] ] ]) ] ][128X[104X
    [4X[25Xgap>[125X [27XExtremelyStrongShodaPairs(SymmetricGroup(5));[127X[104X
    [4X[28X[ [ Sym( [ 1 .. 5 ] ), Sym( [ 1 .. 5 ] ) ], [128X[104X
    [4X[28X  [ Sym( [ 1 .. 5 ] ), Alt( [ 1 .. 5 ] ) ] ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X3.2 [33X[0;0YComputing strong Shoda pairs[133X[101X
  
  [1X3.2-1 StrongShodaPairs[101X
  
  [33X[1;0Y[29X[2XStrongShodaPairs[102X( [3XG[103X ) [32X attribute[133X
  [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 [27Xssp:=StrongShodaPairs( SymmetricGroup(4) );;[127X[104X
    [4X[25Xgap>[125X [27XLength(ssp);[127X[104X
    [4X[28X5[128X[104X
    [4X[25Xgap>[125X [27XList(ssp,x->List(x,StructureDescription));[127X[104X
    [4X[28X[ [ "S4", "S4" ], [ "S4", "A4" ], [ "A4", "C2 x C2" ], [ "D8", "C2 x C2" ], [128X[104X
    [4X[28X  [ "D8", "C4" ] ][128X[104X
    [4X[25Xgap>[125X [27Xssp:=StrongShodaPairs( DihedralGroup(64) );;[127X[104X
    [4X[25Xgap>[125X [27XLength(ssp);[127X[104X
    [4X[28X8[128X[104X
    [4X[25Xgap>[125X [27XList(ssp,x->List(x,StructureDescription));[127X[104X
    [4X[28X[ [ "D64", "D64" ], [ "D64", "D32" ], [ "D64", "C32" ], [ "D64", "D32" ], [128X[104X
    [4X[28X  [ "D32", "D16" ], [ "C32", "C4" ], [ "C32", "C2" ], [ "C32", "1" ] ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X3.3 [33X[0;0YProperties related with Shoda pairs[133X[101X
  
  [1X3.3-1 IsExtremelyStrongShodaPair[101X
  
  [33X[1;0Y[29X[2XIsExtremelyStrongShodaPair[102X( [3XG[103X, [3XK[103X, [3XH[103X ) [32X operation[133X
  
  [33X[0;0YThe  first  argument  should  be  a  finite group [3XG[103X, the second one a normal
  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 an [13Xextremely strong Shoda pair[113X ([14X9.16[114X) of [3XG[103X, and
  [9Xfalse[109X otherwise.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XG:=SymmetricGroup(4);; K:=Group( (1,3,2,4), (3,4) );;[127X[104X
    [4X[25Xgap>[125X [27XH1:=Group( (2,4,3), (1,4)(2,3), (1,3)(2,4) );;[127X[104X
    [4X[25Xgap>[125X [27XH2:=Group( (3,4), (1,2)(3,4) );;[127X[104X
    [4X[25Xgap>[125X [27XIsExtremelyStrongShodaPair( G, G, H1 );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsExtremelyStrongShodaPair( G, K, H2 );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsExtremelyStrongShodaPair( G, G, H2 );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsExtremelyStrongShodaPair( G, G, K );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X3.3-2 IsStrongShodaPair[101X
  
  [33X[1;0Y[29X[2XIsStrongShodaPair[102X( [3XG[103X, [3XK[103X, [3XH[103X ) [32X operation[133X
  
  [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
  
  [33X[0;0YNote  that every extremely strong Shoda pair is a strong Shoda pair, but the
  converse is not true.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XG:=SymmetricGroup(4);; K:=Group( (1,3,2,4), (3,4) );;[127X[104X
    [4X[25Xgap>[125X [27XH1:=Group( (2,4,3), (1,4)(2,3), (1,3)(2,4) );;[127X[104X
    [4X[25Xgap>[125X [27XH2:=Group( (3,4), (1,2)(3,4) );;[127X[104X
    [4X[25Xgap>[125X [27XIsStrongShodaPair( G, G, H1 );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsExtremelyStrongShodaPair( G, K, H2 );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsStrongShodaPair( G, K, H2 );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsStrongShodaPair( G, G, K );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X3.3-3 IsShodaPair[101X
  
  [33X[1;0Y[29X[2XIsShodaPair[102X( [3XG[103X, [3XK[103X, [3XH[103X ) [32X operation[133X
  
  [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
  
  [1X3.3-4 IsStronglyMonomial[101X
  
  [33X[1;0Y[29X[2XIsStronglyMonomial[102X( [3XG[103X ) [32X operation[133X
  
  [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 ([14X9.17[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
  
  [1X3.3-5 IsNormallyMonomial[101X
  
  [33X[1;0Y[29X[2XIsNormallyMonomial[102X( [3XG[103X ) [32X operation[133X
  
  [33X[0;0YThe input [3XG[103X should be a finite group.[133X
  
  [33X[0;0YReturns [9Xtrue[109X if [3XG[103X is a finite [13Xnormally monomial[113X ([14X9.18[114X) group.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27X D24:=DihedralGroup(24);[127X[104X
    [4X[28X<pc group of size 24 with 4 generators>[128X[104X
    [4X[25Xgap>[125X [27XIsNormallyMonomial(D24);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XG:=SmallGroup(192,1023);[127X[104X
    [4X[28X<pc group of size 192 with 7 generators>[128X[104X
    [4X[25Xgap>[125X [27XIsNormallyMonomial(G);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XG:=SmallGroup(1029,12); [127X[104X
    [4X[28X<pc group of size 1029 with 4 generators>[128X[104X
    [4X[25Xgap>[125X [27XIsNormallyMonomial(G);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsStronglyMonomial(G);  [127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XG:=SL(2,3);            [127X[104X
    [4X[28XSL(2,3)[128X[104X
    [4X[25Xgap>[125X [27XIsNormallyMonomial(G);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsStronglyMonomial(G);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
