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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 Generalized dihedral group ： ウィキペディア英語版 Generalized dihedral group In mathematics, the generalized dihedral groups are a family of groups with algebraic structures similar to that of the dihedral groups. They include the finite dihedral groups, the infinite dihedral group, and the orthogonal group ''O''(2). ==Definition== For any abelian group ''H'', the generalized dihedral group of ''H'', written Dih(''H''), is the semidirect product of ''H'' and Z2, with Z2 acting on ''H'' by inverting elements. I.e., $\backslash mathrm(H)\; =\; H\; \backslash rtimes\_\backslash phi\; Z\_2$ with φ(0) the identity and φ(1) inversion. Thus we get: :(''h''1, 0) * (''h''2, ''t''2) = (''h''1 + ''h''2, ''t''2) :(''h''1, 1) * (''h''2, ''t''2) = (''h''1 − ''h''2, 1 + ''t''2) for all ''h''1, ''h''2 in ''H'' and ''t''2 in Z2. (Writing Z2 multiplicatively, we have (''h''1, ''t''1) * (''h''2, ''t''2) = (''h''1 + ''t''1''h''2, ''t''1''t''2) .) Note that (''h'', 0) * (0,1) = (''h'',1), i.e. first the inversion and then the operation in ''H''. Also (0, 1) * (''h'', ''t'') = (−''h'', 1 + ''t''); indeed (0,1) inverts ''h'', and toggles ''t'' between "normal" (0) and "inverted" (1) (this combined operation is its own inverse). The subgroup of Dih(''H'') of elements (''h'', 0) is a normal subgroup of index 2, isomorphic to ''H'', while the elements (''h'', 1) are all their own inverse. The conjugacy classes are: *the sets *the sets Thus for every subgroup ''M'' of ''H'', the corresponding set of elements (''m'',0) is also a normal subgroup. We have: ::Dih(''H'') ''/'' ''M'' = Dih ( ''H / M'' ) 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「Generalized dihedral 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.