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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 Primitive permutation group ： ウィキペディア英語版 Primitive permutation group In mathematics, a permutation group ''G'' acting on a set ''X'' is called primitive if ''G'' acts transitively on ''X'' and ''G'' preserves no nontrivial partition of ''X''. Otherwise, if ''G'' is transitive and ''G'' does preserve a nontrivial partition, ''G'' is called imprimitive. This terminology was introduced by Évariste Galois in his last letter, in which he used the French term ''équation primitive'' for an equation whose Galois group is primitive.〔 Galois' last letter: http://www.galois.ihp.fr/ressources/vie-et-oeuvre-de-galois/lettres/lettre-testament〕 In the same letter he stated also the following theorem. If ''G'' is a primitive solvable group acting on a finite set ''X'', then the order of ''X'' is a power of a prime number ''p'', ''X'' may be identified with an affine space over the finite field with ''p'' elements and ''G'' acts on ''X'' as a subgroup of the affine group. An imprimitive permutation group is an example of an induced representation; examples include coset representations ''G''/''H'' in cases where ''H'' is not a maximal subgroup. When ''H'' is maximal, the coset representation is primitive. If the set ''X'' is finite, its cardinality is called the "degree" of ''G''. The numbers of primitive groups of small degree were stated by Robert Carmichael in 1937: Note the large number of primitive groups of degree 16. As Carmichael notes, all of these groups, except for the symmetric and alternating group, are subgroups of the affine group on the 4-dimensional space over the 2-element finite field. While primitive permutation groups are transitive by definition, not all transitive permutation groups are primitive. The requirement that a primitive group be transitive is necessary only when ''X'' is a 2-element set; otherwise, the condition that ''G'' preserves no nontrivial partition implies that ''G'' is transitive. == Examples == * Consider the symmetric group $S\_3$ acting on the set $X=\backslash$ and the permutation : $\backslash eta=\backslash begin$ 1 & 2 & 3 \\ 2 & 3 & 1 \end. Both $S\_3$ and the group generated by $\backslash eta$ are primitive. * Now consider the symmetric group $S\_4$ acting on the set $\backslash$ and the permutation : $\backslash sigma=\backslash begin$ 1 & 2 & 3 & 4 \\ 2 & 3 & 4 & 1 \end. The group generated by $\backslash sigma$ is not primitive, since the partition $(X\_1,\; X\_2)$ where $X\_1\; =\; \backslash$ and $X\_2\; =\; \backslash$ is preserved under $\backslash sigma$, i.e. $\backslash sigma(X\_1)\; =\; X\_2$ and $\backslash sigma(X\_2)=X\_1$. * Every transitive group of prime degree is primitive * The symmetric group $S\_n$ acting on the set $\backslash$ is primitive for every ''n'' and the alternating group $A\_n$ acting on the set $\backslash$ is primitive for every ''n'' > 2. 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「Primitive permutation group」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.