# symmetric group

Let $X$ be a set.
Let $\mathrm{Sym}(X)$ be the set of permutations^{} of $X$
(i.e. the set of bijective functions from $X$ to itself).
Then the act of taking the composition^{} of two permutations
induces a group structure^{} on $\mathrm{Sym}(X)$.
We call this group the symmetric group^{}.

The group $\mathrm{Sym}(\{1,2,\mathrm{\dots},n\})$ is often denoted ${S}_{n}$ or ${\U0001d516}_{n}$.

${S}_{n}$ is generated by the transpositions^{} $\{(1,2),(2,3),\mathrm{\dots},(n-1,n)\}$,
and by any pair of a 2-cycle and $n$-cycle.

${S}_{n}$ is the Weyl group of the ${A}_{n-1}$ root system (and hence of the special linear group^{} $S{L}_{n-1}$).

Title | symmetric group |
---|---|

Canonical name | SymmetricGroup |

Date of creation | 2013-03-22 12:01:53 |

Last modified on | 2013-03-22 12:01:53 |

Owner | bwebste (988) |

Last modified by | bwebste (988) |

Numerical id | 11 |

Author | bwebste (988) |

Entry type | Definition |

Classification | msc 20B30 |

Related topic | Group |

Related topic | Cycle2 |

Related topic | CayleyGraphOfS_3 |

Related topic | Symmetry2 |