PermutationReplace[expr, perm]
replaces each part in expr by its image under the permutation perm.

PermutationReplace[expr, gr]
returns the list of images of expr under all elements of the permutation group gr.


  • For an integer in expr present in the cycles of the permutation perm, the image is the integer to the right of , or the first integer of the cycle if is the last one. For an integer not present in the cycles of perm, the image is itself.
  • If g is a permutation object in expr, then the action is understood as right conjugation: PermutationProduct[InversePermutation[perm], g, perm]. This is equivalent to replacing the points in the cycles of g by their images under perm.
  • When applied to a permutation group expr, PermutationReplace operates on each individual generator, returning the same abstract group but acting on different points.
  • Both arguments are independently listable. If both arguments are lists then the second argument is threaded first.

ExamplesExamplesopen allclose all

Basic Examples (2)Basic Examples (2)

The image of integer 4 under Cycles[{{2, 3, 4, 6}}] is integer 6:

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Under the identity, permutation integers are not moved:

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An action of a permutation on another permutation is understood as conjugation:

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Images under all elements of a group:

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