(***********************************************************************

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(*CacheID: 232*)


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Notebook[{

Cell[CellGroupData[{
Cell[TextData[{
  "An automorphism of ",
  Cell[BoxData[
      \(TraditionalForm\`S\_6\)]],
  " that is not an inner automorphism"
}], "Title"],

Cell["Bernd Fiedler, Leipzig, October 2000", "Subtitle"],

Cell["\<\
Bernd Fiedler, Alfred-Rosch-Str. 13, D-04249 Leipzig, Germany
Bernd.Fiedler.RoschStr.Leipzig@t-online.de\
\>", "Subsubtitle"],

Cell[TextData[{
  "The symmetric group ",
  Cell[BoxData[
      \(TraditionalForm\`S\_6\)]],
  " is the only one that possesses an automorphism that is not an inner \
automorphism.\n\nLiteratur:  Kurzweil, Hans and Stellmacher, Bernd, Theorie \
der endlichen Gruppen, Springer-Verlag, Berlin, Heidelberg, New York et al., \
1998, p. 89, proplems 16, 17. See also p.88."
}], "Text"],

Cell[CellGroupData[{

Cell["<<perms`perms`", "Input"],

Cell["\<\
------------------------------------------------------------
  PERMS, Version 2.2, (05.07.2000), for Mathematica 2.x-4.0
  Copyright (c) 1994-2000 Bernd Fiedler
  A.-Rosch-Str. 13, Leipzig, Germany
  email: Bernd.Fiedler.RoschStr.Leipzig@t-online.de
------------------------------------------------------------

-- For formatted output in a Version 3 or 4 notebook:
   Cell menu  -> Default Intput Format Type -> InputForm
              -> Default Output Format Type -> OutputForm
 
Enter the PERMS configuration which is intended to load.
-------------------------------------------------------------
 (m) Minimal configuration with character tables of S1...S10
 (v) Full version with character tables of S1...S17
 
The evaluation of CHARTAB.M is running. Please wait.\
\>", "Print"]
}, Open  ]],

Cell[TextData[{
  "The ",
  Cell[BoxData[
      \(TraditionalForm\`S\_6\)]],
  "can be generated by two generators a = {2, 3, 4, 5, 1, 6} and b = {6, 2, \
3, 4, 5, 1} which have the orders 5 and 2, respectively. The types are \
type(a) = (0,0,0,0,1,0) and type(b) = (4,1,0,0,0,0)."
}], "Text"],

Cell[CellGroupData[{

Cell["\<\
gens = HoldList[Perm[2,3,4,5,1,6],Perm[6,2,3,4,5,1]]
\
\>", "Input"],

Cell[OutputFormData["\<\
HoldList[Perm[2, 3, 4, 5, 1, 6], Perm[6, 2, 3, 4, 5, 1]]\
\>", "\<\
{( 2 3 4 5 1 6 ), ( 6 2 3 4 5 1 )}\
\>"], "Output"]
}, Open  ]],

Cell["grp1 = GeneratedGroup[gens];", "Input"],

Cell[CellGroupData[{

Cell["Length[grp1]", "Input"],

Cell[OutputFormData["\<\
720\
\>", "\<\
720\
\>"], "Output"]
}, Open  ]],

Cell[TextData[{
  "Now we search for other generators of ",
  Cell[BoxData[
      \(TraditionalForm\`S\_6\)]],
  " with orders 5 and 2 which have other types than a or b."
}], "Text"],

Cell["ordersets = OrderSets[grp1];", "Input"],

Cell[CellGroupData[{

Cell["orders = Table[ordersets[[i,1]], {i,Length[ordersets]}]", "Input"],

Cell[OutputFormData["\<\
{6, 5, 4, 3, 2, 1}\
\>", "\<\
{6, 5, 4, 3, 2, 1}\
\>"], "Output"]
}, Open  ]],

Cell[TextData[{
  "Thus the elements of ",
  StyleBox["grp1",
    FontWeight->"Bold"],
  " of order5 are"
}], "Text"],

Cell[CellGroupData[{

Cell["order5elements = ordersets[[2,2]]", "Input"],

Cell[OutputFormData["\<\
HoldList[Perm[1, 3, 4, 5, 6, 2], Perm[1, 3, 4, 6, 2, 5], 
  Perm[1, 3, 5, 2, 6, 4], Perm[1, 3, 5, 6, 4, 2], Perm[1, 3, 6, 2, 4, 5], 
  Perm[1, 3, 6, 5, 2, 4], Perm[1, 4, 2, 5, 6, 3], Perm[1, 4, 2, 6, 3, 5], 
  Perm[1, 4, 5, 3, 6, 2], Perm[1, 4, 5, 6, 2, 3], Perm[1, 4, 6, 3, 2, 5], 
  Perm[1, 4, 6, 5, 3, 2], Perm[1, 5, 2, 3, 6, 4], Perm[1, 5, 2, 6, 4, 3], 
  Perm[1, 5, 4, 2, 6, 3], Perm[1, 5, 4, 6, 3, 2], Perm[1, 5, 6, 2, 3, 4], 
  Perm[1, 5, 6, 3, 4, 2], Perm[1, 6, 2, 3, 4, 5], Perm[1, 6, 2, 5, 3, 4], 
  Perm[1, 6, 4, 2, 3, 5], Perm[1, 6, 4, 5, 2, 3], Perm[1, 6, 5, 2, 4, 3], 
  Perm[1, 6, 5, 3, 2, 4], Perm[2, 3, 4, 5, 1, 6], Perm[2, 3, 4, 6, 5, 1], 
  Perm[2, 3, 5, 1, 4, 6], Perm[2, 3, 5, 4, 6, 1], Perm[2, 3, 6, 1, 5, 4], 
  Perm[2, 3, 6, 4, 1, 5], Perm[2, 4, 1, 5, 3, 6], Perm[2, 4, 1, 6, 5, 3], 
  Perm[2, 4, 3, 5, 6, 1], Perm[2, 4, 3, 6, 1, 5], Perm[2, 4, 5, 3, 1, 6], 
  Perm[2, 4, 6, 3, 5, 1], Perm[2, 5, 1, 3, 4, 6], Perm[2, 5, 1, 4, 6, 3], 
  Perm[2, 5, 3, 1, 6, 4], Perm[2, 5, 3, 6, 4, 1], Perm[2, 5, 4, 1, 3, 6], 
  Perm[2, 5, 6, 4, 3, 1], Perm[2, 6, 1, 3, 5, 4], Perm[2, 6, 1, 4, 3, 5], 
  Perm[2, 6, 3, 1, 4, 5], Perm[2, 6, 3, 5, 1, 4], Perm[2, 6, 4, 1, 5, 3], 
  Perm[2, 6, 5, 4, 1, 3], Perm[3, 1, 4, 5, 2, 6], Perm[3, 1, 4, 6, 5, 2], 
  Perm[3, 1, 5, 2, 4, 6], Perm[3, 1, 5, 4, 6, 2], Perm[3, 1, 6, 2, 5, 4], 
  Perm[3, 1, 6, 4, 2, 5], Perm[3, 2, 4, 5, 6, 1], Perm[3, 2, 4, 6, 1, 5], 
  Perm[3, 2, 5, 1, 6, 4], Perm[3, 2, 5, 6, 4, 1], Perm[3, 2, 6, 1, 4, 5], 
  Perm[3, 2, 6, 5, 1, 4], Perm[3, 4, 2, 5, 1, 6], Perm[3, 4, 2, 6, 5, 1], 
  Perm[3, 4, 5, 1, 2, 6], Perm[3, 4, 6, 1, 5, 2], Perm[3, 5, 2, 1, 4, 6], 
  Perm[3, 5, 2, 4, 6, 1], Perm[3, 5, 4, 2, 1, 6], Perm[3, 5, 6, 4, 1, 2], 
  Perm[3, 6, 2, 1, 5, 4], Perm[3, 6, 2, 4, 1, 5], Perm[3, 6, 4, 2, 5, 1], 
  Perm[3, 6, 5, 4, 2, 1], Perm[4, 1, 2, 5, 3, 6], Perm[4, 1, 2, 6, 5, 3], 
  Perm[4, 1, 3, 5, 6, 2], Perm[4, 1, 3, 6, 2, 5], Perm[4, 1, 5, 3, 2, 6], 
  Perm[4, 1, 6, 3, 5, 2], Perm[4, 2, 1, 5, 6, 3], Perm[4, 2, 1, 6, 3, 5], 
  Perm[4, 2, 5, 3, 6, 1], Perm[4, 2, 5, 6, 1, 3], Perm[4, 2, 6, 3, 1, 5], 
  Perm[4, 2, 6, 5, 3, 1], Perm[4, 3, 1, 5, 2, 6], Perm[4, 3, 1, 6, 5, 2], 
  Perm[4, 3, 5, 2, 1, 6], Perm[4, 3, 6, 2, 5, 1], Perm[4, 5, 1, 2, 3, 6], 
  Perm[4, 5, 2, 3, 1, 6], Perm[4, 5, 3, 2, 6, 1], Perm[4, 5, 3, 6, 1, 2], 
  Perm[4, 6, 1, 2, 5, 3], Perm[4, 6, 2, 3, 5, 1], Perm[4, 6, 3, 2, 1, 5], 
  Perm[4, 6, 3, 5, 2, 1], Perm[5, 1, 2, 3, 4, 6], Perm[5, 1, 2, 4, 6, 3], 
  Perm[5, 1, 3, 2, 6, 4], Perm[5, 1, 3, 6, 4, 2], Perm[5, 1, 4, 2, 3, 6], 
  Perm[5, 1, 6, 4, 3, 2], Perm[5, 2, 1, 3, 6, 4], Perm[5, 2, 1, 6, 4, 3], 
  Perm[5, 2, 4, 1, 6, 3], Perm[5, 2, 4, 6, 3, 1], Perm[5, 2, 6, 1, 3, 4], 
  Perm[5, 2, 6, 3, 4, 1], Perm[5, 3, 1, 2, 4, 6], Perm[5, 3, 1, 4, 6, 2], 
  Perm[5, 3, 4, 1, 2, 6], Perm[5, 3, 6, 4, 2, 1], Perm[5, 4, 1, 3, 2, 6], 
  Perm[5, 4, 2, 1, 3, 6], Perm[5, 4, 3, 1, 6, 2], Perm[5, 4, 3, 6, 2, 1], 
  Perm[5, 6, 1, 4, 2, 3], Perm[5, 6, 2, 4, 3, 1], Perm[5, 6, 3, 1, 2, 4], 
  Perm[5, 6, 3, 2, 4, 1], Perm[6, 1, 2, 3, 5, 4], Perm[6, 1, 2, 4, 3, 5], 
  Perm[6, 1, 3, 2, 4, 5], Perm[6, 1, 3, 5, 2, 4], Perm[6, 1, 4, 2, 5, 3], 
  Perm[6, 1, 5, 4, 2, 3], Perm[6, 2, 1, 3, 4, 5], Perm[6, 2, 1, 5, 3, 4], 
  Perm[6, 2, 4, 1, 3, 5], Perm[6, 2, 4, 5, 1, 3], Perm[6, 2, 5, 1, 4, 3], 
  Perm[6, 2, 5, 3, 1, 4], Perm[6, 3, 1, 2, 5, 4], Perm[6, 3, 1, 4, 2, 5], 
  Perm[6, 3, 4, 1, 5, 2], Perm[6, 3, 5, 4, 1, 2], Perm[6, 4, 1, 3, 5, 2], 
  Perm[6, 4, 2, 1, 5, 3], Perm[6, 4, 3, 1, 2, 5], Perm[6, 4, 3, 5, 1, 2], 
  Perm[6, 5, 1, 4, 3, 2], Perm[6, 5, 2, 4, 1, 3], Perm[6, 5, 3, 1, 4, 2], 
  Perm[6, 5, 3, 2, 1, 4]]\
\>", "\<\
{( 1 3 4 5 6 2 ), ( 1 3 4 6 2 5 ), ( 1 3 5 2 6 4 ), ( 1 3 5 6 4 2 ), 
 
  ( 1 3 6 2 4 5 ), ( 1 3 6 5 2 4 ), ( 1 4 2 5 6 3 ), ( 1 4 2 6 3 5 ), 
 
  ( 1 4 5 3 6 2 ), ( 1 4 5 6 2 3 ), ( 1 4 6 3 2 5 ), ( 1 4 6 5 3 2 ), 
 
  ( 1 5 2 3 6 4 ), ( 1 5 2 6 4 3 ), ( 1 5 4 2 6 3 ), ( 1 5 4 6 3 2 ), 
 
  ( 1 5 6 2 3 4 ), ( 1 5 6 3 4 2 ), ( 1 6 2 3 4 5 ), ( 1 6 2 5 3 4 ), 
 
  ( 1 6 4 2 3 5 ), ( 1 6 4 5 2 3 ), ( 1 6 5 2 4 3 ), ( 1 6 5 3 2 4 ), 
 
  ( 2 3 4 5 1 6 ), ( 2 3 4 6 5 1 ), ( 2 3 5 1 4 6 ), ( 2 3 5 4 6 1 ), 
 
  ( 2 3 6 1 5 4 ), ( 2 3 6 4 1 5 ), ( 2 4 1 5 3 6 ), ( 2 4 1 6 5 3 ), 
 
  ( 2 4 3 5 6 1 ), ( 2 4 3 6 1 5 ), ( 2 4 5 3 1 6 ), ( 2 4 6 3 5 1 ), 
 
  ( 2 5 1 3 4 6 ), ( 2 5 1 4 6 3 ), ( 2 5 3 1 6 4 ), ( 2 5 3 6 4 1 ), 
 
  ( 2 5 4 1 3 6 ), ( 2 5 6 4 3 1 ), ( 2 6 1 3 5 4 ), ( 2 6 1 4 3 5 ), 
 
  ( 2 6 3 1 4 5 ), ( 2 6 3 5 1 4 ), ( 2 6 4 1 5 3 ), ( 2 6 5 4 1 3 ), 
 
  ( 3 1 4 5 2 6 ), ( 3 1 4 6 5 2 ), ( 3 1 5 2 4 6 ), ( 3 1 5 4 6 2 ), 
 
  ( 3 1 6 2 5 4 ), ( 3 1 6 4 2 5 ), ( 3 2 4 5 6 1 ), ( 3 2 4 6 1 5 ), 
 
  ( 3 2 5 1 6 4 ), ( 3 2 5 6 4 1 ), ( 3 2 6 1 4 5 ), ( 3 2 6 5 1 4 ), 
 
  ( 3 4 2 5 1 6 ), ( 3 4 2 6 5 1 ), ( 3 4 5 1 2 6 ), ( 3 4 6 1 5 2 ), 
 
  ( 3 5 2 1 4 6 ), ( 3 5 2 4 6 1 ), ( 3 5 4 2 1 6 ), ( 3 5 6 4 1 2 ), 
 
  ( 3 6 2 1 5 4 ), ( 3 6 2 4 1 5 ), ( 3 6 4 2 5 1 ), ( 3 6 5 4 2 1 ), 
 
  ( 4 1 2 5 3 6 ), ( 4 1 2 6 5 3 ), ( 4 1 3 5 6 2 ), ( 4 1 3 6 2 5 ), 
 
  ( 4 1 5 3 2 6 ), ( 4 1 6 3 5 2 ), ( 4 2 1 5 6 3 ), ( 4 2 1 6 3 5 ), 
 
  ( 4 2 5 3 6 1 ), ( 4 2 5 6 1 3 ), ( 4 2 6 3 1 5 ), ( 4 2 6 5 3 1 ), 
 
  ( 4 3 1 5 2 6 ), ( 4 3 1 6 5 2 ), ( 4 3 5 2 1 6 ), ( 4 3 6 2 5 1 ), 
 
  ( 4 5 1 2 3 6 ), ( 4 5 2 3 1 6 ), ( 4 5 3 2 6 1 ), ( 4 5 3 6 1 2 ), 
 
  ( 4 6 1 2 5 3 ), ( 4 6 2 3 5 1 ), ( 4 6 3 2 1 5 ), ( 4 6 3 5 2 1 ), 
 
  ( 5 1 2 3 4 6 ), ( 5 1 2 4 6 3 ), ( 5 1 3 2 6 4 ), ( 5 1 3 6 4 2 ), 
 
  ( 5 1 4 2 3 6 ), ( 5 1 6 4 3 2 ), ( 5 2 1 3 6 4 ), ( 5 2 1 6 4 3 ), 
 
  ( 5 2 4 1 6 3 ), ( 5 2 4 6 3 1 ), ( 5 2 6 1 3 4 ), ( 5 2 6 3 4 1 ), 
 
  ( 5 3 1 2 4 6 ), ( 5 3 1 4 6 2 ), ( 5 3 4 1 2 6 ), ( 5 3 6 4 2 1 ), 
 
  ( 5 4 1 3 2 6 ), ( 5 4 2 1 3 6 ), ( 5 4 3 1 6 2 ), ( 5 4 3 6 2 1 ), 
 
  ( 5 6 1 4 2 3 ), ( 5 6 2 4 3 1 ), ( 5 6 3 1 2 4 ), ( 5 6 3 2 4 1 ), 
 
  ( 6 1 2 3 5 4 ), ( 6 1 2 4 3 5 ), ( 6 1 3 2 4 5 ), ( 6 1 3 5 2 4 ), 
 
  ( 6 1 4 2 5 3 ), ( 6 1 5 4 2 3 ), ( 6 2 1 3 4 5 ), ( 6 2 1 5 3 4 ), 
 
  ( 6 2 4 1 3 5 ), ( 6 2 4 5 1 3 ), ( 6 2 5 1 4 3 ), ( 6 2 5 3 1 4 ), 
 
  ( 6 3 1 2 5 4 ), ( 6 3 1 4 2 5 ), ( 6 3 4 1 5 2 ), ( 6 3 5 4 1 2 ), 
 
  ( 6 4 1 3 5 2 ), ( 6 4 2 1 5 3 ), ( 6 4 3 1 2 5 ), ( 6 4 3 5 1 2 ), 
 
  ( 6 5 1 4 3 2 ), ( 6 5 2 4 1 3 ), ( 6 5 3 1 4 2 ), ( 6 5 3 2 1 4 )}\
\>"], "Output"]
}, Open  ]],

Cell["\<\
All these permutations have a cycle partition (5 1) or a type \
(0,0,0,0,1,0):\
\>", "Text"],

Cell[CellGroupData[{

Cell["Union[Perm2Part[#]& /@ order5elements]", "Input"],

Cell[OutputFormData["\<\
HoldList[Parti[5, 1]]\
\>", "\<\
{{5, 1}}\
\>"], "Output"]
}, Open  ]],

Cell[TextData[{
  "Thus the permutations of ",
  StyleBox["order5elements",
    FontWeight->"Bold"],
  " ly in the same conjugacy class and we can restrict us to a = gens[[1]]. a \
is an even permutation:"
}], "Text"],

Cell[CellGroupData[{

Cell["Signature[gens[[1]]]", "Input"],

Cell[OutputFormData["\<\
1\
\>", "\<\
1\
\>"], "Output"]
}, Open  ]],

Cell[TextData[{
  "Permutation of order 2 have a cycle partition (2 ",
  Cell[BoxData[
      \(TraditionalForm\`1\^4\)]],
  "), (",
  Cell[BoxData[
      \(TraditionalForm\`2\^2\)]],
  " ",
  Cell[BoxData[
      \(TraditionalForm\`1\^2\)]],
  ") or (",
  Cell[BoxData[
      \(TraditionalForm\`2\^3\)]],
  "). We search for a new generator b' of order 2.\n\nIf b' has a cycle \
partition (2 ",
  Cell[BoxData[
      \(TraditionalForm\`1\^4\)]],
  ") then the mapping grp1 = \[LeftAngleBracket]a,b\[RightAngleBracket] \
\[Rule] grp2 = \[LeftAngleBracket]a,b'\[RightAngleBracket] with a \
\[RightTeeArrow] a and b \[RightTeeArrow] b' is an inner automorphism of ",
  Cell[BoxData[
      \(TraditionalForm\`S\_6\)]],
  ".\n",
  StyleBox["Proof:",
    FontWeight->"Bold"],
  " Since a = (1 2 3 4 5) = {2,3,4,5,1,6}, the transposition b' cannot have \
the fixed point 6, i.e. b' = (k 6), k \[Element]{1,2,3,4,5}. If we set q := ",
  
  Cell[BoxData[
      \(TraditionalForm\`a\^\(k - 1\)\)]],
  ", then q(1) = k. Furthermore, it follows q\[CenterDot]a \[CenterDot]",
  Cell[BoxData[
      \(TraditionalForm\`q\^\(-1\)\)]],
  " = a and  q\[CenterDot]b \[CenterDot]",
  Cell[BoxData[
      \(TraditionalForm\`q\^\(-1\)\)]],
  " = b'  \[DoubleRightArrow] \[LeftAngleBracket]a,b'\[RightAngleBracket] = ",
  
  Cell[BoxData[
      \(TraditionalForm\`q\^\(-1\)\)]],
  "\[LeftAngleBracket]a,b\[RightAngleBracket] q .\n\nExample:"
}], "Text"],

Cell[CellGroupData[{

Cell["a = gens[[1]]", "Input"],

Cell[OutputFormData["\<\
Perm[2, 3, 4, 5, 1, 6]\
\>", "\<\
( 2 3 4 5 1 6 )\
\>"], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell["q = PermPower[a,2]", "Input"],

Cell[OutputFormData["\<\
Perm[3, 4, 5, 1, 2, 6]\
\>", "\<\
( 3 4 5 1 2 6 )\
\>"], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell["PermProd[q,PermProd[a,InvPerm[q]]]", "Input"],

Cell[OutputFormData["\<\
Perm[2, 3, 4, 5, 1, 6]\
\>", "\<\
( 2 3 4 5 1 6 )\
\>"], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell["b = gens[[2]]", "Input"],

Cell[OutputFormData["\<\
Perm[6, 2, 3, 4, 5, 1]\
\>", "\<\
( 6 2 3 4 5 1 )\
\>"], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell["PermProd[q,PermProd[b,InvPerm[q]]]", "Input"],

Cell[OutputFormData["\<\
Perm[1, 2, 6, 4, 5, 3]\
\>", "\<\
( 1 2 6 4 5 3 )\
\>"], "Output"]
}, Open  ]],

Cell[TextData[{
  "If b' has a cycle partition (",
  Cell[BoxData[
      \(TraditionalForm\`2\^2\)]],
  " ",
  Cell[BoxData[
      \(TraditionalForm\`1\^2\)]],
  "). Thus b' is also an even permutation and grp2 = \[LeftAngleBracket]a,b'\
\[RightAngleBracket] is a subgroup of ",
  Cell[BoxData[
      \(TraditionalForm\`A\_6\)]],
  ".\n\nFinally, ",
  Cell[BoxData[
      \(TraditionalForm\`S\_6\)]],
  "contains exactly the following 15 permutations with a cycle partition (",
  Cell[BoxData[
      \(TraditionalForm\`2\^3\)]],
  "):\n\np1 = (1 2)(3 4)(5 6)     p6 = (1 3)(2 6)(4 5)    p11 = (1 5)(2 4)(3 \
6)\np2 = (1 2)(3 5)(4 6)     p7 = (1 4)(2 3)(5 6)    p12 = (1 5)(2 6)(3 4)\n\
p3 = (1 2)(3 6)(4 5)     p8 = (1 4)(2 5)(3 6)    p13 = (1 6)(2 3)(4 5)\np4 = \
(1 3)(2 4)(5 6)     p9 = (1 4)(2 6)(3 5)    p14 = (1 6)(2 4)(3 5)\np5 = (1 \
3)(2 5)(4 6)   p10 = (1 5)(2 3)(4 6)    p15 = (1 6)(2 5)(3 4)\n\nWe determine \
these permutations by means of thePERMS-function Perm2Part:"
}], "Text"],

Cell[CellGroupData[{

Cell["newb = Select[grp1, Perm2Part[#] === Parti[2,2,2]&]", "Input"],

Cell[OutputFormData["\<\
HoldList[Perm[3, 5, 1, 6, 2, 4], Perm[4, 5, 6, 1, 2, 3], 
  Perm[5, 6, 4, 3, 1, 2], Perm[6, 3, 2, 5, 4, 1], Perm[2, 1, 4, 3, 6, 5], 
  Perm[4, 6, 5, 1, 3, 2], Perm[6, 4, 5, 2, 3, 1], Perm[3, 4, 1, 2, 6, 5], 
  Perm[5, 3, 2, 6, 1, 4], Perm[2, 1, 6, 5, 4, 3], Perm[2, 1, 5, 6, 3, 4], 
  Perm[5, 4, 6, 2, 1, 3], Perm[4, 3, 2, 1, 6, 5], Perm[3, 6, 1, 5, 4, 2], 
  Perm[6, 5, 4, 3, 2, 1]]\
\>", "\<\
{( 3 5 1 6 2 4 ), ( 4 5 6 1 2 3 ), ( 5 6 4 3 1 2 ), ( 6 3 2 5 4 1 ), 
 
  ( 2 1 4 3 6 5 ), ( 4 6 5 1 3 2 ), ( 6 4 5 2 3 1 ), ( 3 4 1 2 6 5 ), 
 
  ( 5 3 2 6 1 4 ), ( 2 1 6 5 4 3 ), ( 2 1 5 6 3 4 ), ( 5 4 6 2 1 3 ), 
 
  ( 4 3 2 1 6 5 ), ( 3 6 1 5 4 2 ), ( 6 5 4 3 2 1 )}\
\>"], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Length[newb]", "Input"],

Cell[OutputFormData["\<\
15\
\>", "\<\
15\
\>"], "Output"]
}, Open  ]],

Cell[TextData[
"Now we determine the orders of the groups \[LeftAngleBracket]a,b'\
\[RightAngleBracket] with b' \[Element] newb."], "Text"],

Cell[CellGroupData[{

Cell["grp2orders = Length[GeneratedGroup[{gens[[1]],#}]]& /@ newb", "Input"],

Cell[OutputFormData["\<\
HoldList[120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 720, 720, 720, 
  720, 720]\
\>", "\<\
{120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 720, 720, 720, 720, 720}\
\>"], "Output"]
}, Open  ]],

Cell[TextData[{
  "We see that \[LeftAngleBracket]a,b'\[RightAngleBracket] = ",
  Cell[BoxData[
      \(TraditionalForm\`S\_6\)]],
  " if b' is one of the last 5 elements of the list newb. In all other cases \
we have |\[LeftAngleBracket]a,b\[RightAngleBracket]'| = 120. The groups \
\[LeftAngleBracket]a,b'\[RightAngleBracket] with |\[LeftAngleBracket]a,b\
\[RightAngleBracket]'| = 120 are isomorphic to ",
  Cell[BoxData[
      \(TraditionalForm\`S\_5\)]],
  "= {p \[Element] ",
  Cell[BoxData[
      \(TraditionalForm\`S\_6\)]],
  " | p(6) = 6 }. However these \[LeftAngleBracket]a,b'\[RightAngleBracket] \
are not conjugate to ",
  Cell[BoxData[
      \(TraditionalForm\`S\_5\)]],
  " since every such \[LeftAngleBracket]a,b'\[RightAngleBracket] contains \
elements b' that have no fixed points."
}], "Text"]
}, Open  ]]
},
FrontEndVersion->"Microsoft Windows 3.0",
ScreenRectangle->{{0, 1024}, {0, 712}},
WindowToolbars->"EditBar",
WindowSize->{594, 555},
WindowMargins->{{6, Automatic}, {Automatic, 5}}
]


(***********************************************************************
Cached data follows.  If you edit this Notebook file directly, not using
Mathematica, you must remove the line containing CacheID at the top of 
the file.  The cache data will then be recreated when you save this file 
from within Mathematica.
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Cell[CellGroupData[{
Cell[1731, 51, 143, 5, 150, "Title"],
Cell[1877, 58, 56, 0, 64, "Subtitle"],
Cell[1936, 60, 135, 3, 71, "Subsubtitle"],
Cell[2074, 65, 381, 8, 90, "Text"],

Cell[CellGroupData[{
Cell[2480, 77, 31, 0, 30, "Input"],
Cell[2514, 79, 794, 18, 297, "Print"]
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*)




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End of Mathematica Notebook file.
***********************************************************************)