Editing Symmetric group (section)

== Generators and relations ==
The symmetric group on {{mvar|n}} letters is generated by the [[adjacent transposition]]s <math> \sigma_i = (i, i + 1)</math> that swap {{mvar|i}} and {{math|''i'' + 1}}.<ref>{{citation|title = The Symmetric Group| edition = 2 | first = Bruce E. | last = Sagan | author-link = Bruce Sagan | publisher = Springer | year = 2001 | page = [{{GBurl|Jm-HBaMdt8sC|p=4}} 4] |isbn=978-0-387-95067-9}}</ref>  The collection <math>\sigma_1, \ldots, \sigma_{n-1}</math> generates {{math|S<sub>''n''</sub>}} subject to the following relations:<ref>{{citation|title = Combinatorics of Coxeter groups|last1 = Björner|first1 = Anders|author1-link = Anders Björner|last2 = Brenti|first2 = Francesco|publisher = Springer|year = 2005 |page = [{{GBurl|1TBPz5sd8m0C|p=4}} 4. Example 1.2.3] |isbn=978-3-540-27596-1}}</ref>
*<math>\sigma_i^2 = 1,</math>
*<math>\sigma_i\sigma_j = \sigma_j\sigma_i</math> for <math>|i-j| > 1</math>, and
*<math>(\sigma_i\sigma_{i+1})^3 =1,</math>
where 1 represents the identity permutation.  This representation endows the symmetric group with the structure of a [[Coxeter group]] (and so also a [[reflection group]]).

Other possible generating sets include the set of transpositions that swap {{math|1}} and {{mvar|i}} for {{math|2 ≤ ''i'' ≤ ''n''}},<ref>{{citation|title = Minimal factorizations of permutations into star transpositions | author1 = J. Irving | author2 = A. Rattan | journal = Discrete Math. | volume = 309 | year = 2009 | issue = 6 | pages = 1435–1442 | doi = 10.1016/j.disc.2008.02.018| hdl = 1721.1/96203 | hdl-access = free }}</ref> or more generally any set of transpositions that forms a connected graph,<ref>{{citation| title = Hurwitz Numbers for Reflection Groups I: Generatingfunctionology | author1 = Theo Douvropoulos | author2 = Joel Brewster Lewis | author3 = Alejandro H. Morales | journal = Enumerative Combinatorics and Applications | issue = 3 | volume = 2 | year = 2022 | doi = 10.54550/ECA2022V2S3R20 | at = Proposition 2.1| arxiv = 2112.03427 }}</ref> and a set containing any {{mvar|n}}-cycle and a {{math|2}}-cycle of adjacent elements in the {{mvar|n}}-cycle.<ref>{{citation|title = Algebra | first = Michael | last = Artin | author-link = Michael Artin | publisher = Pearson | year = 1991 | at = Exercise 6.6.16 |isbn=978-0-13-004763-2}}</ref><ref>{{citation|title = Short presentations for alternating and symmetric groups| first1 = J.N. | last1 = Bray | first2 = M.D.E. | last2 = Conder | first3 = C.R.| last3 = Leedham-Green| first4 = E.A. | last4 = O'Brien | publisher = Transactions of the AMS | year = 2007}}</ref>