# cycle notation

The cycle notation is a useful convention for writing down a
permutations^{} in terms of its constituent cycles. Let $S$ be a finite
set^{}, and

$${a}_{1},\mathrm{\dots},{a}_{k},k\ge 2$$ |

distinct elements of $S$. The expression $({a}_{1},\mathrm{\dots},{a}_{k})$ denotes the cycle whose action is

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

Note there are $k$ different expressions for the same cycle; the following all represent the same cycle:

$$({a}_{1},{a}_{2},{a}_{3},\mathrm{\dots},{a}_{k})=({a}_{2},{a}_{3},\mathrm{\dots},{a}_{k},{a}_{1}),=\mathrm{\dots}=({a}_{k},{a}_{1},{a}_{2},\mathrm{\dots},{a}_{k-1}).$$ |

Also note that a 1-element cycle is
the same thing as the identity^{} permutation, and thus there is not
much point in writing down such things. Rather, it is customary to
express the identity permutation simply as $()$ or $(1)$.

Let $\pi $ be a permutation of $S$, and let

$${S}_{1},\mathrm{\dots},{S}_{k}\subset S,k\in \mathbb{N}$$ |

be the orbits of $\pi $ with more than 1 element. For each $j=1,\mathrm{\dots},k$ let ${n}_{j}$ denote the cardinality of ${S}_{j}$. Also, choose an ${a}_{1,j}\in {S}_{j}$, and define

$${a}_{i+1,j}=\pi ({a}_{i,j}),i\in \mathbb{N}.$$ |

We can now express $\pi $ as a product^{} of disjoint cycles, namely

$$\pi =({a}_{1,1},\mathrm{\dots}{a}_{{n}_{1},1})({a}_{2,1},\mathrm{\dots},{a}_{{n}_{2},2})\mathrm{\dots}({a}_{k,1},\mathrm{\dots},{a}_{{n}_{k},k}).$$ |

By way of illustration, here are the 24 elements of the symmetric
group^{} on $\{1,2,3,4\}$ expressed using the cycle notation, and grouped
according to their conjugacy classes^{}:

$(),$ | ||

$(12),(13),(14),(23),(24),(34)$ | ||

$(123),(213),(124),(214),(134),(143),(234),(243)$ | ||

$(12)(34),(13)(24),(14)(23)$ | ||

$(1234),(1243),(1324),(1342),(1423),(1432)$ |

Title | cycle notation |
---|---|

Canonical name | CycleNotation |

Date of creation | 2013-03-22 12:33:41 |

Last modified on | 2013-03-22 12:33:41 |

Owner | rmilson (146) |

Last modified by | rmilson (146) |

Numerical id | 6 |

Author | rmilson (146) |

Entry type | Definition |

Classification | msc 20B05 |

Classification | msc 05A05 |

Related topic | Cycle2 |

Related topic | Permutation |

Related topic | OneLineNotationForPermutations |