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SECTION 8 Groups of Permutations Definition A permutation of a set A is a function ϕ: AA that is both one to one and onto. If and are both permutations of a set A, then the composite function A A gives a one-to-one and onto defined by A mapping of A into A. We can show that function composition is a binary operation, and call this function composition permutation multiplication. We will denote by . Remember that the action of on A must be read in right-to-left order: first apply and then . Notations Example: Suppose A = {1, 2, 3, 4, 5} and that is the permutation given by 1 4, 2 2, 3 5, 4 3, 5 1. We can write as following: 1 2 3 4 5 4 2 5 3 1 , then = Let 3 5 4 2 1 12345 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 4 2 5 3 1 3 5 4 2 1 5 1 3 2 4 Permutation Groups Theorem Le A be a nonempty set, and let SA be the collection of all permutations of A. Then SA is a group under permutation multiplication. Proof: exercise. Symmetric Groups Note: here we will focus on the case where A is finite. it’s also customary to take A to be set of the form {1, 2, 3, …, , n} for some positive integer n. Definition: Let A be the finite set {1, 2, , n}. The group of all permutations of A is the symmetric group on n letters , and is denoted by Sn. Note that Sn has n! elements, where n!=n(n-1)(n-2) (3)(2)(1). Two important examples Example: S3 Let set A be {1, 2, 3}. Then S3 is a group with 3!=6 elements. Let 1 2 3 1 , 1 1 1 2 3 1 2 3 1 1 , 2 3 2 3 1 1 2 3 1 2 , 3 2 3 1 2 0 2 3 , 3 2 2 3 , 2 1 2 3 . 1 3 Then the multiplication table for S3 is shown in the next slide. 0 1 2 1 2 0 0 1 2 1 2 3 3 1 1 2 0 3 1 2 2 2 0 1 2 3 1 1 1 2 3 0 1 2 2 2 3 1 2 0 1 3 3 1 2 1 2 0 S3 and D3 Note that this group is not abelian ( 1u1 u11 ) There is a natural correspondence between the elements of S3 and the ways in which two copies of an equilateral triangle with vertices 1, 2, and 3 can be placed, one covering the other with vertices on to of vertices. For this reason, S3 is also the group D3 of symmetries of an equilateral triangle. Naively, we used i for rotations and i for mirror images in bisectors of angles. 3 1 2 Cayley’s Theorem Definition Let f: A B be a function and let H be a subset of A. The image of H under f is { f (h) | h H } and is denoted by f [H]. Lemma Let G and G’ be groups and let : G G’ be a one-to-one function such that (x y) = (x) (y) for all x, y G. Then [G] is a subgroup of G’ and provides an isomorphism of G with [G]. Then apply the above Lemma, we can show Theorem (Cayley’s Theorem) Every group is isomorphic to a group of permutations.