# 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

 $a_{1},\ldots,a_{k},\quad k\geq 2$

distinct elements of $S$. The expression $(a_{1},\ldots,a_{k})$ denotes the cycle whose action is

 $a_{1}\mapsto a_{2}\mapsto a_{3}\ldots a_{k}\mapsto a_{1}.$

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

 $(a_{1},a_{2},a_{3},\ldots,a_{k})=(a_{2},a_{3},\ldots,a_{k},a_{1}),=\ldots=(a_{% k},a_{1},a_{2},\ldots,a_{k-1}).$

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

 $S_{1},\ldots,S_{k}\subset S,\quad k\in\mathbb{N}$

be the orbits of $\pi$ with more than 1 element. For each $j=1,\ldots,k$ let $n_{j}$ denote the cardinality of $S_{j}$. Also, choose an $a_{1,j}\in S_{j}$, and define

 $a_{i+1,j}=\pi(a_{i,j}),\quad i\in\mathbb{N}.$

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

 $\pi=(a_{1,1},\ldots a_{n_{1},1})(a_{2,1},\ldots,a_{n_{2},2})\ldots(a_{k,1},% \ldots,a_{n_{k},k}).$

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:

 $\displaystyle(),$ $\displaystyle(12),\;(13),\;(14),\;(23),\;(24),\;(34)$ $\displaystyle(123),\;(213),\;(124),\;(214),\;(134),\;(143),\;(234),\;(243)$ $\displaystyle(12)(34),\;(13)(24),\;(14)(23)$ $\displaystyle(1234),\;(1243),\;(1324),\;(1342),\;(1423),\;(1432)$
Title cycle notation CycleNotation 2013-03-22 12:33:41 2013-03-22 12:33:41 rmilson (146) rmilson (146) 6 rmilson (146) Definition msc 20B05 msc 05A05 Cycle2 Permutation OneLineNotationForPermutations