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| {{about||the cyclic decomposition of [[Graph theory|graphs]]|Cycle decomposition (graph theory)|[[cycling]] terminology|glossary of bicycling}}
| | The author's title is Christy. For years she's been residing in Kentucky but her spouse wants them to move. What me and my family adore is bungee leaping but I've been taking on new things recently. For years she's been working as a travel agent.<br><br>My blog; [http://appin.co.kr/board_Zqtv22/688025 psychics online] |
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| In [[combinatorics|combinatorial]] [[mathematics]], the '''cycle notation''' is a useful convention for writing down a [[permutation]] in terms of its constituent [[cycle (mathematics)|cycle]]s.<ref>Fraleigh 2002:89; Hungerford 1997:230</ref> This is also called '''circular notation''' and the permutation called a '''cyclic''' or '''circular''' permutation.<ref>Dehn 1930:19</ref>
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| ==Definition==
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| Let <math>S</math> be the set <math>\{1,\dots,n\}, n \in \mathbb{N}</math>, and
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| :<math> a_1,\ldots,a_k,\quad 1 \leq k \leq n</math>
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| be distinct elements of <math>S</math>. The expression
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| :<math>(a_1\ \ldots\ a_k)</math>
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| denotes the cycle σ whose [[Group action|action]] is
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| :<math> a_1\mapsto a_2\mapsto a_3\mapsto \ldots \mapsto a_k \mapsto a_1.</math>
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| For each index ''i'', | |
| :<math>\sigma (a_i) = a_{i+1},</math>
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| where <math>a_{k+1}</math> is taken to mean <math>a_1</math>.
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| There are <math>k</math> different expressions for the same cycle; the following all represent the same cycle:
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| :<math> (a_1\ a_2\ a_3\ \ldots\ a_k) = (a_2\ a_3\ \ldots\ a_k\ a_1) = \cdots = (a_k\ a_1\ a_2\ \ldots\ a_{k-1}).\, </math>
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| A 1-element cycle such as (3) is the [[identity function|identity]] permutation.<ref>Hungerford 1997:231</ref> The identity permutation can also be written as an empty cycle, "()".<ref>Johnson 2003:691</ref>
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| ==Permutation as product of cycles==
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| Let <math>\pi</math> be a permutation of <math>S</math>, and let
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| :<math> S_1,\ldots, S_k\subset S,\quad k\in\mathbb{N}</math>
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| be the [[Orbit (group theory)#Orbits and stabilizers|orbit]]s of <math>\pi</math> with more than 1 element. Consider an element <math>S_j</math>, <math>j=1,\ldots,k</math>, let <math>n_j</math> denote the cardinality of <math>S_j</math>,<math>|S_j|</math> =<math>n_j</math>. Also, choose an <math>a_{1,j}\in S_j</math>, and define
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| :<math> a_{i+1,j} = \pi(a_{i,j}),\quad\text{for } 1\leq i<n_j;\quad\text{then also }\pi(a_{n_j,j})=a_{1,j}.\,</math>
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| We can now express <math>\pi</math> as a product of disjoint cycles, namely
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| :<math> \pi = (a_{1,1}\ \ldots a_{n_1,1}) (a_{1,2}\ \ldots\ a_{n_2,2}) \ldots (a_{1,k}\ \ldots\ a_{n_k,k}).\,</math>
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| Since disjoint cycles commute with each other, the meaning of this expression is independent of the convention used for the order in products of permutations, namely whether the factors in such a product operate rightmost-first (as is usual more generally for [[function composition]]), or leftmost-first as some authors prefer. The meaning of individual cycles is also independent of this convention, namely always as described above.
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| ==Example==
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| Here are the 24 elements of the [[symmetric group]] on <math>\{1,2,3,4\}</math> expressed using the cycle notation, and grouped according to their [[conjugacy class]]es:
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| ::<math> ( )\,</math>
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| ::<math> (1 2), \;(1 3),\; (1 4),\; (2 3),\; (2 4),\; (3 4)</math> ([[Transposition (mathematics)|transpositions]])
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| ::<math> (1 2 3),\; (1 3 2),\; (1 2 4),\; (1 4 2),\; (1 3 4),\; (1 4 3),\; (2 3 4),\; (2 4 3)</math> | |
| ::<math> (1 2)(3 4),\;(1 3)(2 4),\; (1 4)(2 3)</math>
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| ::<math> (1 2 3 4),\; (1 2 4 3),\; (1 3 2 4),\; (1 3 4 2),\; (1 4 2 3),\; (1 4 3 2)</math>
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| ==See also==
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| *[[Cyclic permutation]]
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| *[[Cycles and fixed points]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *{{citation|title=Algebraic Equations |first=Edgar |last=Dehn|publisher=Dover |year=1960 |origyear=1930}}.
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| *{{citation|first=John|last=Fraleigh|authorlink=|title=A first course in abstract algebra|edition=7th|year=2003|publisher=Addison Wesley|isbn=978-0-201-76390-4|page=88–90}}.
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| *{{citation|first=Thomas W.|last=Hungerford|title=Abstract Algebra: An Introduction|year=1997|publisher=Brooks/Cole|isbn=978-0-03-010559-3}}.
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| *{{citation|first=James L.|last=Johnson|title=Probability and Statistics for Computer Science|year=2003|publisher=Wiley Interscience|isbn=978-0-471-32672-4}}.
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| {{PlanetMath attribution|id=2808|title=cycle notation}}
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| [[Category:Permutations]]
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The author's title is Christy. For years she's been residing in Kentucky but her spouse wants them to move. What me and my family adore is bungee leaping but I've been taking on new things recently. For years she's been working as a travel agent.
My blog; psychics online