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I want to make an action table of the symmetric group (say $S_4$) acting on the set of unordered pairs of distinct vertices of a square. I have the list of such pairs:

Level[Table[Table[{i, j}, {i, j + 1, 4}], {j, 1, 3}], {2}]

I want to map each element of this list to the unordered pair corresponding to a vertex permutation of $S_4$. For example: the pair {3,2} under the action of {2,1,4,3} = Cycles[{{1,2},{3,4}] would correspond to the pair {4,1}.

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    $\begingroup$ PermutationReplace may be the function you're looking for. The output of your example can be obtained for instance with PermutationReplace[{3, 2}, Cycles[{{1, 2}, {3, 4}}]]. $\endgroup$ – user31159 Oct 17 '15 at 19:29
  • $\begingroup$ Yes! Thank you very much. $\endgroup$ – Geoffrey Critzer Oct 17 '15 at 19:37
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Making an answer from my comment.


Notation

list = Level[Table[Table[{i, j}, {i, j + 1, 4}], {j, 1, 3}], {2}]

(* {{2, 1}, {3, 1}, {4, 1}, {3, 2}, {4, 2}, {4, 3}} *)

Proposition

PermutationReplace[list, Cycles[{{1, 2}, {3, 4}}]]

(* {{1, 2}, {4, 2}, {3, 2}, {4, 1}, {3, 1}, {3, 4}} *)
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