SECTION 8 Groups of Permutations

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SECTION 8 Groups of Permutations

• Definition
• A permutation of a set A is a function ?? A?A
  that is both one to one and onto.
• If ??and ? are both permutations of a set A, then
  the composite function
• ? ?? defined by
  gives a one-to-one and onto 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 ?.

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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

Let , then ???
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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.

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Symmetric Groups

• Note here we will focus on the case where A is
  finite. its 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).

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Two important examples

• Example S3
• Let set A be 1, 2, 3. Then S3 is a group with
  3!6 elements. Let
• Then the multiplication table for S3 is shown in
  the next slide.

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S3 and D3

• Note that this group is not abelian (
  )
• 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.

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Cayleys 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 (Cayleys Theorem)
• Every group is isomorphic to a group of
  permutations.