# 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:

Under the identity, permutation integers are not moved:

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

Images under all elements of a group:

## Scope(6)

The image of a point in the support of the permutation is the right neighbor of the point:

The image of the last point of a cycle is the first point of that cycle:

A point not present in the permutation support stays invariant:

PermutationReplace on arrays of integers returns the list of respective images:

PermutationReplace on other permutations is understood as conjugation:

On a permutation group, the generators are conjugated:

The second argument is listable:

If both arguments are lists then the second argument is threaded first:

Images under all elements of a group:

## Properties & Relations(4)

PermutationReplace is a right action with respect to PermutationProduct:

PermutationReplace on an identity permutation list coincides with PermutationList:

PermutationReplace on an identity permutation list produces the inverse result of Permute:

The orbit of a point under a permutation group is the union of the images of that point under the elements of the group:

## Neat Examples(1)

Graphical representation of the elements of groups by sorted lists of images: