6.3 Permutation groups and cyclic groups

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6.3 Permutation groups and cyclic groups

• Example Consider the equilateral triangle with
  vertices 1,2,and 3. Let l1, l2, and l3 be the
  angle bisectors of the corresponding angles, and
  let O be their point of intersection?
• Counterclockwise rotation of the triangle about O
  through 120,240,360 (0)

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• f21?2,2?3,3?1
• f31?3,2?1,3?2
• f1 1?1,2?2,3?3
• reflect the lines l1, l2, and l3.
• g11?1,2?3,3?2
• g21?3,2?2,3?1
• g31?2,2?1,3?3

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• 6.3.1 Permutation groups
• Definition 9 A bijection from a set S to itself
  is called a permutation of S
• Lemma 6.1Let S be a set.
• (1) Let f and g be two permutations of S. Then
  the composition of f and g is a permutation of S.
• (2) Let f be a permutation of S. Then the
  inverse of f is a permutation of S.

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• Theorem 6.9Let S be a set. The set of all
  permutations of S, under the operation of
  composition of permutations, forms a group A(S).
• Proof Lemma 6.1 implies that the rule of
  multiplication is well-defined.
• associative.
• the identity function from S to S is identity
  element
• The inverse permutation g of f is a permutation
  of S

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• Theorem 6.10 Let S be a finite set with n
  elements. Then A(S) has n! elements.
• Definition 10 The group Sn is the set of
  permutations of the first n natural numbers. The
  group is called the symmetric group on n letters,
  is called also the permutation group.

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• Definition 11 Let Sn, and let ??Sn.We say
  that ? is a d-cycle if there are integers i1 i2
  id such that ?(i1) i2, ?(i2) i3, , and
  ?(id) i1 and ? fixes every other integer, i.e.

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• ?(i1,, id)
• A 2-cycle ? is called transposition.
• Theorem 6.11. Let ? be any element of Sn. Then ?
  may be expressed as a product of disjoint cycles.
• Corollary 6.1. Every permutation of Sn is a
  product of transpositions.

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• Theorem 6.12 If a permutation of Sn can be
  written as a product of an even number of
  transpositions, then it can never be written as a
  product of an odd number of transpositions, and
  conversely.
• Definition 12A permutation of Sn is called even
  it can be written as a product of an even number
  of transpositions, and a permutation of Sn is
  called odd if it can never be written as a
  product of an odd number of transpositions.

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• (i1 i2 ik)(i1 i2)(i2 i3)(ik-2 ik-1)(ik-1 ik)
• k-1

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• ? Even permutation Odd
• Even permutation Even permutation Odd
• Odd permutation Odd permutation Even

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• ? Even permutation odd
  permutation
• Even permutation Even permutation
  Odd permutation
• Odd permutation Odd permutation
  Even permutation
• Sn On?An
• OnnAn?
• An? is a group?

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• Theorem 6.13 The set of even permutations forms
  a group, is called the altemating group of degree
  n and denoted by An. The order of An is n!/2(
  where ngt1)
• An?
• n1,An1?
• ngt1,
• AnOnn!/2

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• Next cyclic groups, Subgroups, Normal subgroups
• Exercise
• P212 (Sixth) OR P195(Fifth) 8,9, 12,15,21