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Permutations

Definition

Definition: let X be a set.

  • A bijection of X to itself is called a permutation of X. The set of all permutations of X is denoted by \text{Sym}(X) and is called the symmetric group on X.
  • The product g \cdot h of two permutations g,h in \text{Sym}(X) is defined as the composition g \circ h of g and h.
  • If X = \{1, \dots, n\} we write \mathrm{Sym}_n(X) instead of \mathrm{Sym}(X).

Definition: the identity map is defined as \mathrm{id}: X \to X with g = g \cdot \mathrm{id} = \mathrm{id} \cdot g for all g in \mathrm{Sym}(X). The inverse of g denoted by g^{-1} satisfies g^{-1} \cdot g = g \cdot g^{-1} = \mathrm{id}.

In matrix notation: let g = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 3 & 1\end{pmatrix} and h = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 1 & 3\end{pmatrix} with g,h \in \mathrm{Sym}_3(X), then we can take

g \cdot h = \begin{pmatrix} 1 & 2 & 3 \ 2 & 3 & 1 \ \hline 2 & 1 & 3 \ 3 & 2 & 1\end{pmatrix} = \begin{pmatrix} 1 & 2 & 3 \ 3 & 2 & 1\end{pmatrix},

and we have g^{-1} = \begin{pmatrix} 2 & 3 & 1 \\1 & 2 & 3 \end{pmatrix}.


Theorem: \mathrm{Sym}_n has exactly n! elements.

??? note "Proof:"

A permutation can be described in a matrix notation by a $2$ by $n$ matrix with the numbers $1,\dots,n$ in the first row and the images in the second row. There are $n!$ possibilities to fill the second row.

We can also omit the matrix notation and use the list notation for permutations then we have for g = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 3 & 1\end{pmatrix} = [2,3,1], as the first row speaks for itself.


Definition: the order of a permutation g is the smallest positive integer m such that g^m = \mathrm{id}.

For example the order of the permutation [2,1,3] in \mathrm{Sym}_3 is 2.

If g is a permutation in \mathrm{Sym}_n then the permutations g, g^2, g^3, \dots can not all be distinct, since there are only n! distinct permutations in \mathrm{Sym}_n. So there must exists a r < s such that g^r = g^s. Since g is a bijection there must be g^{s-r} = e. So there exist positive numbers m with g^m = e and in particular a smallest such number. Therefore each permutation g has a well-defined order.