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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 Retract (group theory) ： ウィキペディア英語版 Retract (group theory) In mathematics, in the field of group theory, a subgroup of a group is termed a retract if there is an endomorphism of the group that maps surjectively to the subgroup and is identity on the subgroup. In symbols, $H$ is a retract of $G$ if and only if there is an endomorphism $\backslash sigma:G\; \backslash to\; G$ such that $\backslash sigma(h)\; =\; h$ for all $h\; \backslash in\; H$ and $\backslash sigma(g)\; \backslash in\; H$ for all $g\; \backslash in\; G$.〔〔.〕 The endomorphism itself (having this property) is an idempotent element in the transformation monoid of endomorphisms, so it called an idempotent endomorphism〔.〕〔.〕 or a retraction.〔 The following is known about retracts: * A subgroup is a retract if and only if it has a normal complement.〔.〕 The normal complement, specifically, is the kernel of the retraction. * Every direct factor is a retract.〔 Conversely, any retract which is a normal subgroup is a direct factor.〔For an example of a normal subgroup that is not a retract, and therefore is not a direct factor, see .〕 * Every retract has the congruence extension property. * Every regular factor, and in particular, every free factor, is a retract. ==References== 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「Retract (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.