cycle notation

The cycle notation is a useful convention for writing down apermutations in terms of its constituent cycles. Let $S$ be a finiteset, and

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

distinct elements of $S$ . Theexpression $(a_{1},\\ldots,a_{k})$ 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; thefollowing 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 isthe same thing as the identity permutation, and thus there is notmuch point in writing down such things. Rather, it is customary toexpress 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 $n_{j}$ denotethe cardinality of $S_{j}$ . Also, choose an $a_{1,j}\\in S_{j}$ , anddefine

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 symmetricgroup on $\\{1,2,3,4\\}$ expressed using the cycle notation, and groupedaccording to their conjugacy classes:

	$\\displaystyle(),$
	$\\displaystyle(12),\\;(13),\\;(14),\\;(23),\\;(24),\\;(34)$
	$\\displaystyle(123),\\;(213),\\;(124),\\;(214),\\;(134),\\;(143),\\;(234),\\;(243)$
	$\\displaystyle(12)(34),\\;(13)(24),\\;(14)(23)$
	$\\displaystyle(1234),\\;(1243),\\;(1324),\\;(1342),\\;(1423),\\;(1432)$