Mo logo [home] [lexicon] [problems] [tests] [courses] [auxiliaries] [notes] [staff] german flag
Mathematics-Online course: Linear Algebra - Basic Structures - Groups and Fields

Transposition and Sign of Permutations

[previous page] [next page] [table of contents][page overview]

A transposition

$\displaystyle \tau = (j,k) $

is an exchange of $ j$ and $ k$. By composition of these elementary permutations, any permutation $ \pi$ can be represented by:

$\displaystyle \pi = \tau_1 \circ \cdots \circ \tau_m\, , $

where the parity (even or odd $ m$) is uniquely determined. Thus, the so called sign of permutation $ \pi$ is well defined by

$\displaystyle \sigma(\pi) = (-1)^m\, . $

(Authors: Burkhardt/Höllig/Knesch)

Let

$ \tau_1\circ\cdots\circ \tau_m = \pi \in S_n, \ k=\pi(n)$,
then

$\displaystyle \tilde \pi = (k,n)\circ \pi \in S_{n-1}\, , $

since $ n$ does not change under the latter permutation. Now assume that the parity of the induced decomposition of $ \tilde \pi$ is uniquely determined. It follows, by induction, that the sign of $ \pi$ is uniquely determined as well and equals $ (-1)^m$.

(Authors: Burkhardt/Höllig/Knesch)
  automatically generated 4/21/2005