Sn is a symmetric group, also called permutation group, of order n!. Notation for permutation:

• cycles: (a1 a2 ...ap)(aq) which means a1 -> a2 -> ... -> ap -> a1 and

  aq -> aq

Theorems:

• Every permutation can be expressed as a product of cycles.
• Two disjoing cycles commute (a1..ap)(b1...bq) ai!=bj for all i and j
• Every finite group of order n is isomorphic to a subgroup of Sn.
• Any permutation can be expressed as a product of 2-cycles.
• An even permutation is the product of even number of 2-cycles.
• An odd permutation is the product of odd number of 2-cycles.
• Two elements of Sn are in the same conjugacy class <=> they have the same cycle structure. i.e. (123)(45) is equivalent to (124)(35) For H subset of Sn, ha~hp=> Sa~Sp but not the converse example: S3~D3

  {e} ~ {(1)(2)(3)} {c,c^2} ~ {(123),(132)} {b,bc,bc^2} ~ {(12)(3), (13)(2),(23)(1)}

C3 subset D3

        {e}
        {c}
        {c^2}