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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 Random group ： ウィキペディア英語版 Random group In mathematics, random groups are certain groups obtained by a probabilistic construction. They were introduced by Misha Gromov to answer questions such as "What does a typical group look like?" It so happens that, once a precise definition is given, random groups satisfy some properties with very high probability, whereas other properties fail with very high probability. For instance, very probably random groups are hyperbolic groups. In this sense, one can say that "most groups are hyperbolic". ==Definition== The definition of random groups depends on a probabilistic model on the set of possible groups. Various such probabilistic models yield different (but related) notions of random groups. Any group can by defined by a group presentation involving generators and relations. For instance, the Abelian group $\backslash mathbb\backslash times\backslash mathbb$ has a presentation with two generators $a$ and $b$, and the relation $ab=ba$, or equivalently $aba^b^=1$. The main idea of random groups is to start with a fixed number of group generators $a\_1,\backslash ,a\_2,\backslash ,\backslash ldots,\backslash ,a\_m$, and imposing relations of the form $r\_1=1,\backslash ,r\_2=1,\backslash ,\backslash ldots,\backslash ,r\_k=1$ where each $r\_j$ is a random word involving the letters $a\_i$ and their formal inverses $a\_i^$. To specify a model of random groups is to specify a precise way in which $m$, $k$ and the random relations $r\_j$ are chosen. Once the random relations $r\_k$ have been chosen, the resulting random group $G$ is defined in the standard way for group presentations, namely: $G$ is the quotient of the free group $F\_m$ with generators $a\_1,\backslash ,a\_2,\backslash ,\backslash ldots,\backslash ,a\_m$, by the normal subgroup $R\backslash subset\; F\_m$ generated by the relations $r\_1\backslash ,r\_2,\backslash ,\backslash ldots,\backslash ,r\_k$ seen as elements of $F\_m$: : $G=F\_m/\backslash langle\; r\_1,\backslash ,r\_2,\backslash ,\backslash ldots,\backslash ,r\_k\; \backslash rangle.$ 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「Random 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.