# cycle

Let $S$ be a set. A *cycle* is a permutation^{}
(bijective function of a set onto itself)
such that there exist distinct elements ${a}_{1},{a}_{2},\mathrm{\dots},{a}_{k}$ of $S$
such that

$$f({a}_{i})={a}_{i+1}\mathit{\hspace{1em}\hspace{1em}}\text{and}\mathit{\hspace{1em}\hspace{1em}}f({a}_{k})={a}_{1}$$ |

that is

$f({a}_{1})$ | $=$ | ${a}_{2}$ | ||

$f({a}_{2})$ | $=$ | ${a}_{3}$ | ||

$\mathrm{\vdots}$ | ||||

$f({a}_{k})$ | $=$ | ${a}_{1}$ |

and $f(x)=x$ for any other element of $S$.

This can also be pictured as

$${a}_{1}\mapsto {a}_{2}\mapsto {a}_{3}\mapsto \mathrm{\cdots}\mapsto {a}_{k}\mapsto {a}_{1}$$ |

and

$$x\mapsto x$$ |

for any other element $x\in S$, where $\mapsto $ represents the action of $f$.

One of the basic results on symmetric groups^{}
says that any finite permutation can be expressed as product^{} of disjoint cycles.

Title | cycle |

Canonical name | Cycle1 |

Date of creation | 2013-03-22 12:24:23 |

Last modified on | 2013-03-22 12:24:23 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 10 |

Author | yark (2760) |

Entry type | Definition |

Classification | msc 03-00 |

Classification | msc 05A05 |

Classification | msc 20F55 |

Related topic | Permutation |

Related topic | SymmetricGroup |

Related topic | Transposition^{} |

Related topic | Group |

Related topic | Subgroup^{} |

Related topic | DihedralGroup |

Related topic | CycleNotation |

Related topic | PermutationNotation |