# PermutationReplace

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.

# Details

• 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.

# Examples

open allclose all

## 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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