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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 Subnormal subgroup ： ウィキペディア英語版 Subnormal subgroup In mathematics, in the field of group theory, a subgroup ''H'' of a given group ''G'' is a subnormal subgroup of ''G'' if there is a finite chain of subgroups of the group, each one normal in the next, beginning at ''H'' and ending at ''G''. In notation, $H$ is $k$-subnormal in $G$ if there are subgroups :$H=H\_0,H\_1,H\_2,\backslash ldots,\; H\_k=G$ of $G$ such that $H\_i$ is normal in $H\_$ for each $i$. A subnormal subgroup is a subgroup that is $k$-subnormal for some positive integer $k$. Some facts about subnormal subgroups: * A 1-subnormal subgroup is a proper normal subgroup (and vice versa). * A finitely generated group is nilpotent if and only if each of its subgroups is subnormal. * Every quasinormal subgroup, and, more generally, every conjugate-permutable subgroup, of a finite group is subnormal. * Every pronormal subgroup that is also subnormal, is normal. In particular, a Sylow subgroup is subnormal if and only if it is normal. * Every 2-subnormal subgroup is a conjugate-permutable subgroup. The property of subnormality is transitive, that is, a subnormal subgroup of a subnormal subgroup is subnormal. The relation of subnormality can be defined as the transitive closure of the relation of normality. If every subnormal subgroup of ''G'' is normal in ''G'', then ''G'' is called a T-group. ==See also== *Characteristic subgroup *Normal core *Normal closure *Ascendant subgroup *Descendant subgroup *Serial subgroup 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「Subnormal subgroup」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.