DG

Permutations

A permutation is a bijection $\sigma$ : {1,...n} -> {1,...n}.
The set of all permutations of {1,...n} is denoted by $\Sigma$ _n.

Permutations $\sigma$ are uniquely determined by distinct k-tuples ( $\sigma$ (1),... $\sigma$ (n)), so | $\Sigma$ _n| = n!

Theorem: $\Sigma$ _n is a group under composition (the symmetric group), which acts on {1,...n}.

Theorem:

Each permutation can be decomposed essentially uniquely into disjoint cycles.
Each permutation can be decomposed into transpositions with unique parity.
Parity is a group homomorphism $\Sigma$ _n -> Z₂
Sign of a permutation, sgn $\sigma$ = (-1)^parity(),
is a group homomorphism sgn $\Sigma$ _n -> exp Z₂ = {-1, 1}.

Proof: Let $\sigma$ $\in$ $\Sigma$ _n. For each k in {1,...n} consider the orbit { $\sigma$ ⁱ(k)}.
The orbits are disjoint, because if $\sigma$ ⁱ(k) = $\sigma$ ^j(m) and i $\ge$ j, then m = $\sigma$ ^i-j(k).
The action of $\sigma$ on each orbit is cyclic, because by the pigeonhole principle
$\exists$ i > j such that $\sigma$ ⁱ(k) = $\sigma$ ^j(k), so k = $\sigma$ ^i-j(k).
Each cycle can be decomposed into transpositions as follows: (1, 2, ... k) = (1, k)...(1, 3)(1, 2).
For each $\sigma$ $\in$ $\Sigma$ _n define $\Delta$ ( $\sigma$ ) = $\prod$ _{i < j} $\sigma$ (i)- $\sigma$ (j). Note that $\Delta$ ( $\sigma$ ) is never 0.
Uniqueness of parity follows, since if $\tau$ is a transposition, then $\Delta$ ( $\sigma$ ) = - $\Delta$ ( $\tau$ $\sigma$ ).

Last updated: Jun 17 18:53 / Last fetched: Tue Dec 2 09:51:45 CST 2008