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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.
MORE INFORMATION
  • 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.
EXAMPLESCLOSE ALL
Basic Examples   (2)
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:
In[1]:=
Click for copyable input
Out[1]=
Under the identity, permutation integers are not moved:
In[2]:=
Click for copyable input
Out[2]=
An action of a permutation on another permutation is understood as conjugation:
In[3]:=
Click for copyable input
Out[3]=
 
Images under all elements of a group:
In[1]:=
Click for copyable input
Out[1]=
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 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:
Neat Examples   (1)
Graphical representation of the elements of groups by sorted lists of images:
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