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# permutation

A *permutation* of a finite set $\{a_{1},\,a_{2},\,\ldots,\,a_{n}\}$ is an arrangement of its elements.
For example, if $S=\{A,\,B,\,C\}$ then $ABC$, $CAB$ , $CBA$ are three different permutations of $S$.

The number of permutations of a set with $n$ elements is $n!$ (see the rule of product).

A permutation can also be seen as a bijective function of a set into itself. For example, the permutation $ABC\mapsto CAB$ could be seen a function $f:\{A,B,C\}\to\{A,B,C\}$ that assigns:

$f(A)=C,\qquad f(B)=A,\qquad f(C)=B.$ |

In fact, every bijection of a set into itself gives a permutation, and any permutation gives rise to a bijective function.

Therefore, we can say that there are $n!$ bijective functions from a set with $n$ elements into itself.

Using the function approach, it can be proved that any permutation can be expressed as a composition of disjoint cycles and also as composition of (not necessarily disjoint) transpositions.

Moreover, if $\sigma=\tau_{1}\tau_{2}\cdots\tau_{m}=\rho_{1}\rho_{2}\cdots\rho_{n}$ are two factorization of a permutation $\sigma$ into transpositions, then $m$ and $n$ must be both even or both odd. So we can label permutations as *even* or *odd* depending on the number of transpositions for any decomposition.

Permutations (as functions) form in general a non-abelian group with function composition as binary operation called *symmetric group of order $n$*. The subset of even permutations becomes a subgroup called the alternating group of order $n$.

## Mathematics Subject Classification

03-00*no label found*20B99

*no label found*46L05

*no label found*82-00

*no label found*83-00

*no label found*81-00

*no label found*22A22

*no label found*05A05

*no label found*

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