# symmetric group

Let $X$ be a set.
Let $S(X)$ be the set of permutations^{} of $X$
(i.e. the set of bijective functions on $X$).
Then the act of taking the composition^{} of two permutations
induces a group structure^{} on $S(X)$.
We call this group the symmetric group^{}
and it is often denoted $\mathrm{Sym}(X)$.

When $X$ has a finite number $n$ of elements, we often refer to the symmetric group as ${S}_{n}$, and describe the elements by using cycle notation.

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

Canonical name | SymmetricGroup1 |

Date of creation | 2013-03-22 14:03:53 |

Last modified on | 2013-03-22 14:03:53 |

Owner | antizeus (11) |

Last modified by | antizeus (11) |

Numerical id | 5 |

Author | antizeus (11) |

Entry type | Definition |

Classification | msc 20B30 |

Related topic | Symmetry2 |