PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (210)
Orphanage
Unclass'd (1)
Unproven (472)
Corrections (38)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: Medium Entry average rating: No information on entry rating
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) ]
(view preamble)

View style:

See Also: cycle, signature of a permutation


Attachments:
signature of a permutation (Definition) by rspuzio
Log in to rate this entry.
(view current ratings)

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 MSC: 03-00 (Mathematical logic and foundations :: General reference works )
 05A05 (Combinatorics :: Enumerative combinatorics :: Combinatorial choice problems )
 20B99 (Group theory and generalizations :: Permutation groups :: Miscellaneous)
Pending Errata and Addenda
None.
[ View all 1 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)