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Symmetric Group, Permutations

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For any set $ M$ the bijections from $ M$ onto $ M$ together with the composition of mappings as operation form a group, the so called symmetric group of $ M$.

If $ M = \{1, 2, \dots, n\}$, then this group is called the symmetric group of degree $ n$ (notation: $ S_n$). The elements of $ S_n$ are called permutations. $ S_n$ has $ n!$ elements. A permutation $ \pi$ can be written as:

$\displaystyle \pi=\left( \begin{array}{ccccc} 1 & 2 & 3 & \dots & n \\ \pi(1) & \pi(2) & \pi(3) & \dots & \pi(n) \end{array} \right) \; . $

The group of permutations is in general not commutative, as shown in the example:

$\displaystyle \left( \begin{array}{ccc} 1 & 2 & 3 \\ 2 & 1 & 3 \end{array}\r... ...) \circ \left( \begin{array}{ccc} 1 & 2 & 3 \\ 2 & 1 & 3 \end{array}\right) $

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[Annotations]
  automatically generated 5/26/2009