Transpositions generate the finitary symmetric group: Difference between revisions
(New page: ==Statement== ===For a finite set=== The symmetric group on a finite set is generated by the transpositions in it. A transposition is a permutation that interchanges two elements...) |
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The [[finitary symmetric group]] on any set is generated by the [[transposition]]s in it. A transposition is a permutation that interchanges two elements and leaves all the others fixed; in other words, it is a cycle of length two. | The [[finitary symmetric group]] on any set is generated by the [[transposition]]s in it. A transposition is a permutation that interchanges two elements and leaves all the others fixed; in other words, it is a cycle of length two. | ||
==Related facts== | |||
* [[3-cycles generate the finitary alternating group]] | |||
* [[Symmetric group on a finite set is 2-generated]] | |||
===In terms of IAPS theory=== | |||
In terms of IAPS theory, this translates to saying that the transposition on a two-element set forms a one-element permutatively generating set for the [[permutation IAPS]]. | |||
==Proof for a finite set== | ==Proof for a finite set== | ||
Revision as of 21:08, 18 August 2008
Statement
For a finite set
The symmetric group on a finite set is generated by the transpositions in it. A transposition is a permutation that interchanges two elements and leaves all the others fixed; in other words, it is a cycle of length two.
For an infinite set
The finitary symmetric group on any set is generated by the transpositions in it. A transposition is a permutation that interchanges two elements and leaves all the others fixed; in other words, it is a cycle of length two.
Related facts
In terms of IAPS theory
In terms of IAPS theory, this translates to saying that the transposition on a two-element set forms a one-element permutatively generating set for the permutation IAPS.