Permutation group

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A permutation group is a mathematical group whose elements are permutations of a given set. For instance, the group of all permutations of a set of n elements is the symmetric group Sn.

For a concrete example, given the set S = {1,2,3,4}, the following permutations form a group:

  • e = (1)(2)(3)(4)

(This is the identity, the trivial permutation which fixes each element.)

  • a = (12)(3)(4)

(This permutation interchanges 1 and 2, and fixes 3 and 4.)

  • b = (1)(2)(34)

(Like the previous one, but exchanging 3 and 4, and fixing the others.)

  • ab = (12)(34)

(This permutation, which is the composition of the previous two, exchanges simultaneously 1 with 2, and 3 with 4.)

Notice that different permutation groups may well be isomorphic as abstract groups. For instance, the permutation group just described is isomorphic (but not equivalent as a permutation group) to {(1)(2)(3)(4), (12)(34), (13)(24), (14)(23)}. Both are isomorphic to the Klein group V4. In fact, every finite group is isomorphic to a permutation group; this is the content of Cayley's theorem.