Cycle (mathematics)

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,ldots,a_k of S such that

Related Topics:
Permutation - Bijective function - Onto

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:f(a_i) = a_{i+1}qquad mbox{and}qquad f(a_k)=a_1

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that is

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:egin{matrix} f(a_1)&=&a_{2}\ f(a_{2})&=&a_{3}\ &dots&\ f(a_{k})&=&a_{1}\ end{matrix}

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

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This can also be pictured as

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:a_1mapsto a_{2}mapsto a_{3}mapstocdotsmapsto a_{k}mapsto a_{1}

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and

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:xmapsto x

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for any other element xin S, where mapsto represents the action of f.

Related Topics:
Represents - Action

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

Related Topics:
Symmetric group - Disjoint

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