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}

source

jl@crush.caltech.edu index

projection_operator

Sn