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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 Absolute presentation of a group ： ウィキペディア英語版 Absolute presentation of a group In mathematics, one method of defining a group is by an absolute presentation.〔B. Neumann, ''The isomorphism problem for algebraically closed groups,'' in: Word Problems, Decision Problems, and the Burnside Problem in Group Theory, Amsterdam-London (1973), pp. 553–562.〕 Recall that to define a group $G\backslash$ by means of a presentation, one specifies a set $S\backslash$ of generators so that every element of the group can be written as a product of some of these generators, and a set $R\backslash$ of relations among those generators. In symbols: :$G\; \backslash simeq\; \backslash langle\; S\; \backslash mid\; R\; \backslash rangle.$ Informally $G\backslash$ is the group generated by the set $S\backslash$ such that $r\; =\; 1\backslash$ for all $r\; \backslash in\; R$. But here there is a tacit assumption that $G\backslash$ is the "freest" such group as clearly the relations are satisfied in any homomorphic image of $G\backslash$. One way of being able to eliminate this tacit assumption is by specifying that certain words in $S\backslash$ should not be equal to $1.$ That is we specify a set $I\backslash$, called the set of irrelations, such that $i\backslash ne\; 1\backslash$ for all $i\; \backslash in\; I$. ==Formal Definition== To define an absolute presentation of a group $G\backslash$ one specifies a set $S\backslash$ of generators, a set $R\backslash$ of relations among those generators and a set $I\backslash$ of irrelations among those generators. We then say $G\backslash$ has absolute presentation :$\backslash langle\; S\; \backslash mid\; R,\; I\backslash rangle.$ provided that: # $G\backslash$ has presentation $\backslash langle\; S\; \backslash mid\; R\backslash rangle.$ # Given any homomorphism $h:G\backslash rightarrow\; H$ such that the irrelations $I\backslash$ are satisfied in $h(G)\backslash$, $G\backslash$ is isomorphic to $h(G)\backslash$. A more algebraic, but equivalent, way of stating condition 2 is: :2a. if $N\backslash triangleleft\; G\backslash$ is a non-trivial normal subgroup of $G$ then $I\backslash cap\; N\backslash neq\; \backslash left\backslash \; .$ Remark: The concept of an absolute presentation has been fruitful in fields such as algebraically closed groups and the Grigorchuk topology. In the literature, in a context where absolute presentations are being discussed, a presentation (in the usual sense of the word) is sometimes referred to as a relative presentation. The term seems rather strange as one may well ask "relative to what?" and the only justification seems to be that ''relative'' is habitually used as an antonym to ''absolute. 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「Absolute presentation of a 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.