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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Transposition (mathematics) ： ウィキペディア英語版 Cyclic permutation In mathematics, and in particular in group theory, a cyclic permutation is a permutation of the elements of some set ''X'' which maps the elements of some subset ''S'' of ''X'' to each other in a cyclic fashion, while fixing (i.e., mapping to themselves) all other elements of ''X''. For example, the permutation of that sends 1 to 3, 3 to 2, 2 to 4 and 4 to 1 is a cycle, while the permutation that sends 1 to 3, 3 to 1, 2 to 4 and 4 to 2 is not (it separately permutes the pairs and ). A cycle in a permutation is a subset of the elements that are permuted in this way. The set ''S'' is called the orbit of the cycle. Every permutation on finitely many elements can be decomposed into a collection of cycles on disjoint orbits. In some contexts, a cyclic permutation itself is called a cycle. == Definition == A permutation is called a cyclic permutation if and only if it consists of a single nontrivial cycle (a cycle of length > 1). Example: :$$ \begin 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ 4 & 2 & 7 & 6 & 5 & 8 & 1 & 3 \end = \begin 1 & 4 & 6 & 8 & 3 & 7 & 2 & 5 \\ 4 & 6 & 8 & 3 & 7 & 1 & 2 & 5 \end = (146837)(2)(5) Some authors restrict the definition to only those permutations which have precisely one cycle (that is, no fixed points allowed). Example: :$$ \begin 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ 4 & 5 & 7 & 6 & 8 & 2 & 1 & 3 \end = \begin 1 & 4 & 6 & 2 & 5 & 8 & 3 & 7 \\ 4 & 6 & 2 & 5 & 8 & 3 & 7 & 1 \end = (14625837) More formally, a permutation of a set ''X'', which is a bijective function $\backslash sigma:X\backslash to\; X$, is called a cycle if the action on ''X'' of the subgroup generated by $\backslash sigma$ has at most one orbit with more than a single element. This notion is most commonly used when ''X'' is a finite set; then of course the largest orbit, ''S'', is also finite. Let $s\_0$ be any element of ''S'', and put $s\_i=\backslash sigma^i(s\_0)\; \backslash ,$ for any $i\backslash in\backslash mathbf$. If ''S'' is finite, there is a minimal number $k\; \backslash geq\; 1$ for which $s\_k=s\_0$. Then $S=\backslash$, and $\backslash sigma$ is the permutation defined by : and $\backslash sigma(x)=x$ for any element of $X\backslash setminus\; S$. The elements not fixed by $\backslash sigma$ can be pictured as :$s\_0\backslash mapsto\; s\_1\backslash mapsto\; s\_2\backslash mapsto\backslash cdots\backslash mapsto\; s\_\backslash mapsto\; s\_k=s\_0$. A cycle can be written using the compact cycle notation $\backslash sigma\; =\; (s\_0~s\_1~\backslash dots~s\_)$ (there are no commas between elements in this notation, to avoid confusion with a ''k''-tuple). The length of a cycle, is the number of elements of its largest orbit. A cycle of length ''k'' is also called a ''k''-cycle. The orbit of a 1-cycle is called a ''fixed point'' of the permutation, but as a permutation every 1-cycle is the identity permutation. When cycle notation is used, the 1-cycles are often suppressed when no confusion will result. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Cyclic permutation」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.