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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 Prüfer group ： ウィキペディア英語版 Prüfer group In mathematics, specifically in group theory, the Prüfer ''p''-group or the ''p''-quasicyclic group or ''p''∞-group, Z(''p''∞), for a prime number ''p'' is the unique ''p''-group in which every element has ''p'' different ''p''-th roots. The Prüfer ''p''-groups are countable abelian groups which are important in the classification of infinite abelian groups: they (along with the group of rational numbers) form the smallest building blocks of all divisible groups. The groups are named after Heinz Prüfer, a German mathematician of the early 20th century. ==Constructions of Z(''p''∞)== The Prüfer ''p''-group may be identified with the subgroup of the circle group, U(1), consisting of all ''p''''n''-th roots of unity as ''n'' ranges over all non-negative integers: :$\backslash mathbf(p^\backslash infty)=\backslash ^+\backslash \}.\backslash ;$ The group operation here is the multiplication of complex numbers. Alternatively and equivalently, the Prüfer ''p''-group may be defined as the Sylow p-subgroup of the quotient group Q''/''Z, consisting of those elements whose order is a power of ''p'': :$\backslash mathbf(p^\backslash infty)\; =\; \backslash mathbf()/\backslash mathbf$ (where Z() denotes the group of all rational numbers whose denominator is a power of ''p'', using addition of rational numbers as group operation). We can also write :$\backslash mathbf(p^\backslash infty)=\backslash mathbf\_p/\backslash mathbf\_p$ where Q''p'' denotes the additive group of ''p''-adic numbers and Z''p'' is the subgroup of ''p''-adic integers. There is a presentation :$\backslash mathbf(p^\backslash infty)\; =\; \backslash langle\backslash ,\; g\_1,\; g\_2,\; g\_3,\; \backslash ldots\; \backslash mid\; g\_1^p\; =\; 1,\; g\_2^p\; =\; g\_1,\; g\_3^p\; =\; g\_2,\; \backslash dots\backslash ,\backslash rangle.$ Here, the group operation in Z(''p''∞) is written as multiplication. 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「Prüfer 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.