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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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III.68 Permutation Groups

Martin W. Liebeck

Let S be a set. A permutation of S is a function from S to S that is both injective and surjective—in other words, a function that “rearranges” the elements of S. For example, if S = {1, 2, 3}, then the function a : SS that sends 1 to 3, 2 to 1, and 3 to 2 is a permutation of S; so is the function b that sends 1 to 3, 2 to 2, and 3 to 1; whereas the function c that sends 1 to 3, 2 to 1, and 3 to 1 is not a permutation. An example of a permutation of the set of real numbers Image is the function x Image 8 - 2 ...

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