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transposition (Definition)

Given a finite set $X=\{a_1,a_2,\ldots,a_n\}$, a transposition is a permutation (bijective function of $X$ onto itself) $f$ such that there exist indices $i,j$ such that $f(a_i)=a_j$, $f(a_j)=a_i$ and $f(a_k)=a_k$ for all other indices $k$. This is often denoted (in the cycle notation) as $(a, b)$.

Example: If $X=\{a,b,c,d,e\}$ the function $\sigma$ given by

\begin{eqnarray*} \sigma(a)&=&a\ \sigma(b)&=&e\ \sigma(c)&=&c\ \sigma(d)&=&d\ \sigma(e)&=&b \end{eqnarray*}


is a transposition.

One of the main results on symmetric groups states that any permutation can be expressed as composition (product) of transpositions, and for any two decompositions of a given permutation, the number of transpositions is always even or always odd.



"transposition" is owned by drini. [ owner history (1) ]
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See Also: cycle, signature of a permutation


Attachments:
signature of a permutation (Definition) by rspuzio
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Cross-references: odd, even, decompositions, product, composition, states, symmetric groups, function, cycle notation, indices, onto, bijective function, permutation, finite set
There are 23 references to this entry.

This is version 3 of transposition, born on 2002-02-20, modified 2004-09-25.
Object id is 2274, canonical name is Transposition.
Accessed 7554 times total.

Classification:
AMS MSC03-00 (Mathematical logic and foundations :: General reference works )
 05A05 (Combinatorics :: Enumerative combinatorics :: Combinatorial choice problems )
 20B99 (Group theory and generalizations :: Permutation groups :: Miscellaneous)

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