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Permutation
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We represent a permutation as two lists of equal length representing preimages
and images of moved points. I.e., fixed points do not occur in either of these
lists. This enables us to compute the set of fixed points and the set of moved
points easily. ::
 
  p := coercePreimagesImages([[1,2,3],[1,2,3]])
    1
                      Type: Permutation PositiveInteger

  movedPoints p
    {}
                      Type: Set PositiveInteger

  even? p
    true
                       Type: Boolean

  p := coercePreimagesImages([[0,1,2,3],[3,0,2,1]])$PERM ZMOD 4
    (1 0 3)
                       Type: Permutation IntegerMod 4

  fixedPoints p
    {2}
                       Type: Set IntegerMod 4

  q := coercePreimagesImages([[0,1,2,3],[1,0]])$PERM ZMOD 4
    (1 0)
                       Type: Permutation IntegerMod 4

  fixedPoints(p*q)
    {2,0}
                       Type: Set IntegerMod 4
  even?(p*q)
    false
                       Type: Boolean

See Also:

* )show Permutation