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Mathematics-Online lexicon:

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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  automatically generated 5/26/2009