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In mathematics, the infinite dihedral group Dih∞ is an infinite group with properties analogous to those of the finite dihedral groups. In two dimensional geometry, the infinite dihedral group represents the 4th frieze group symmetry, ''p1m1'', seen as an infinite set of parallel reflections along an axis. ==Definition== Every dihedral group is generated by a rotation ''r'' and a reflection; if the rotation is a rational multiple of a full rotation, then there is some integer ''n'' such that ''rn'' is the identity, and we have a finite dihedral group of order 2''n''. If the rotation is ''not'' a rational multiple of a full rotation, then there is no such ''n'' and the resulting group has infinitely many elements and is called Dih∞. It has presentations : : and is isomorphic to a semidirect product of Z and Z/2, and to the free product Z/2 * Z/2. It is the automorphism group of the graph consisting of a path infinite to both sides. Correspondingly, it is the isometry group of Z (see also symmetry groups in one dimension), the group of permutations α: Z → Z satisfying |''i'' - ''j''| = |α(''i'') - α(''j'')|, for all ''i, j'' in Z.〔Meenaxi Bhattacharjee, Dugald Macpherson, Rögnvaldur G. Möller, Peter M. Neumann. Notes on Infinite Permutation Groups, Issue 1689. Springer, 1998. (p. 38 ). ISBN 978-3-540-64965-6〕 The infinite dihedral group can also be defined as the holomorph of the infinite cyclic group. 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「Infinite dihedral group」の詳細全文を読む スポンサード リンク
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