翻訳と辞書 Words near each other ・ Holospira oritis ・ Holospira pasonis ・ Holospira pityis ・ Hololena curta ・ Hololepis ・ Hololepta aequalis ・ Hololepta plana ・ Hololoma ・ Holoman-Outland House ・ Holomatix Rendition ・ Holomek ・ Holometabolism ・ Holometer ・ Holomictic lake ・ Holomorph ・ Holomorph (mathematics) ・ Holomorphic curve ・ Holomorphic discrete series representation ・ Holomorphic embedding load flow method ・ Holomorphic function ・ Holomorphic functional calculus ・ Holomorphic Lefschetz fixed-point formula ・ Holomorphic sheaf ・ Holomorphic tangent space ・ Holomorphic vector bundle ・ Holomorphically convex hull ・ Holomorphically separable ・ Holomovement ・ Holomycota ・ Holon Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Holomorph (mathematics) ： ウィキペディア英語版 Holomorph (mathematics) In mathematics, especially in the area of algebra known as group theory, the holomorph of a group is a group which simultaneously contains (copies of) the group and its automorphism group. The holomorph provides interesting examples of groups, and allows one to treat group elements and group automorphisms in a uniform context. In group theory, for a group $G$, the holomorph of $G$ denoted $\backslash operatorname(G)$ can be described as a semidirect product or as a permutation group. ==Hol(''G'') as a semi-direct product== If $\backslash operatorname(G)$ is the automorphism group of $G$ then :$\backslash operatorname(G)=G\backslash rtimes\; \backslash operatorname(G)$ where the multiplication is given by :$(g,\backslash alpha)(h,\backslash beta)=(g\backslash alpha(h),\backslash alpha\backslash beta).$ (1 ) Typically, a semidirect product is given in the form $G\backslash rtimes\_A$ where $G$ and $A$ are groups and $\backslash phi:A\backslash rightarrow\; \backslash operatorname(G)$ is a homomorphism and where the multiplication of elements in the semi-direct product is given as :$(g,a)(h,b)=(g\backslash phi(a)(h),ab)$ which is well defined, since $\backslash phi(a)\backslash in\; \backslash operatorname(G)$ and therefore $\backslash phi(a)(h)\backslash in\; G$. For the holomorph, $A=\backslash operatorname(G)$ and $\backslash phi$ is the identity map, as such we suppress writing $\backslash phi$ explicitly in the multiplication given in (1 ) above. For example, * $G=C\_3=\backslash langle\; x\backslash rangle=\backslash$ the cyclic group of order 3 * $\backslash operatorname(G)=\backslash langle\; \backslash sigma\backslash rangle=\backslash$ where $\backslash sigma(x)=x^2$ * $\backslash operatorname(G)=\backslash$ with the multiplication given by: :$(x^,\backslash sigma^)(x^,\backslash sigma^)\; =\; (x^,\backslash sigma^)$ where the exponents of $x$ are taken mod 3 and those of $\backslash sigma$ mod 2. Observe, for example :$(x,\backslash sigma)(x^2,\backslash sigma)=(x^,\backslash sigma^2)=(x^2,1)$ and note also that this group is not abelian, as $(x^2,\backslash sigma)(x,\backslash sigma)=(x,1)$, so that $\backslash operatorname(C\_3)$ is a non-abelian group of order 6 which, by basic group theory, must be isomorphic to the symmetric group $S\_3$. 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「Holomorph (mathematics)」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.