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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., 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
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