Symmetric group

$Permutation$

σ

is a

n

-permutation:

σ

is bijective

(i_{1}, σ (i_{1}), i_{2}, σ (i_{2}), ... ... i_{n} σ (i_{n})) := σ

n

-permutation

∙

:

\forall σ_{1}, σ_{2} \in S_{n} (∙ (σ_{1}, σ_{2}) = σ_{1} \circ σ_{2})

n

-symmetric group

: (S_{n}, ∙)

σ_{K_{mn}}

is a

m

-cyclic permutation of

Ω

⎩ ⎨ ⎧ σ_{K_{mn}} (k_{1}) = k_{m} σ_{K_{mn}} (k_{i}) = k_{i - 1} σ_{K_{mn}} (p) = p (2 \leq i \leq m) (p \in Ω - K_{m})

(k_{1}, k_{2}, ... k_{m}) := σ_{K_{mn}}

Transposition of

Ω

:

Ω

is

2

-cyclic permutation

All permutation can be expressed as product of cyclic permutations

All cyclic permutation can be expressed as product of transpositions

m_{1}

is even

\Leftrightarrow m_{2}

is even

m_{1}

is odd

\Leftrightarrow m_{2}

is odd

σ

is

n

-permutation:

m_{1}

is even

σ

is

n

-permutation:

m_{1}

is odd

n

- Alternating group:

(A_{n}, ∙)