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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 Complement (group theory) ： ウィキペディア英語版 Complement (group theory) In mathematics, especially in the area of algebra known as group theory, a complement of a subgroup ''H'' in a group ''G'' is a subgroup ''K'' of ''G'' such that :$G\; =\; HK\; =\; \backslash \; \backslash text\; H\backslash cap\; K\; =\; \backslash .$ Equivalently, every element of ''G'' has a unique expression as a product ''hk'' where ''h'' ∈ ''H'' and ''k'' ∈ ''K''. This relation is symmetrical: if ''K'' is a complement of ''H'', then ''H'' is a complement of ''K''. Neither ''H'' nor ''K'' need be a normal subgroup of ''G''. Complements generalize both the direct product (where the subgroups ''H'' and ''K'' commute element-wise), and the semidirect product (where one of ''H'' or ''K'' normalizes the other). The product corresponding to a general complement is called the Zappa–Szép product. In all cases, complement subgroups factor a group into smaller pieces. Some properties of complement subgroups: * Complements need not exist, and if they do they need not be unique. That is, ''H'' could have two distinct complements ''K''1 and ''K''2 in ''G''. * If there are several complements of a normal subgroup, then they are necessarily isomorphic to each other and to the quotient group. * If ''K'' is a complement of ''H'' in ''G'' then ''K'' forms both a left and right transversal of ''H'' (that is, the elements of ''K'' form a complete set of representatives of both the left and right cosets of ''H''). * the Schur–Zassenhaus theorem guarantees the existence of complements of normal Hall subgroups of finite groups. A ''p''-complement is a complement to a Sylow ''p''-subgroup. Theorems of Frobenius and Thompson describe when a group has a normal p-complement. Philip Hall characterized finite soluble groups amongst finite groups as those with ''p''-complements for every prime ''p''; these ''p''-complements are used to form what is called a Sylow system. A Frobenius complement is a special type of complement in a Frobenius group. A complemented group is one where every subgroup has a complement. ==See also== * Product of group subsets 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「Complement (group theory)」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.