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PermutationReplace

PermutationReplace
replaces each part in expr by its image under the permutation perm.
PermutationReplace
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.
The image of integer 4 under Cycles 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:
The image of integer 4 under Cycles is integer 6:
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Out[1]=
Under the identity, permutation integers are not moved:
In[2]:=
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Out[2]=
An action of a permutation on another permutation is understood as conjugation:
In[3]:=
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Out[3]=
 
Images under all elements of a group:
In[1]:=
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Out[1]=
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:
PermutationReplace is a right action with respect to PermutationProduct:
PermutationReplace on lists of integers produces the inverse result of Permute:
PermutationReplace on an identity permutation list is inverse to PermutationList:
The orbit of a point under a permutation group is the union of the images of that point under the elements of the group:
Graphical representation of the elements of groups by sorted lists of images:
New in 8