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When studying the dihedral group of a square, do we consider only vertices or the whole points which the square covers? Because the vertices of square also gives the same symmetries.

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    $\begingroup$ You seem to be answering your own question. The set of vertices of the square has the same symmetries as the whole set of points of the square. Or perhaps I misunderstand your question? $\endgroup$ – Pierre-Guy Plamondon Dec 28 '18 at 14:51
  • $\begingroup$ Absolutely but which set do we consider in dihedral group of order 8 $\endgroup$ – user629838 Dec 28 '18 at 14:53
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When studying the symmetry groups of a whole geometric object, the symmetry of all points of that object is considered.

Take e.g. a square that contains a non-symmetric pattern on its surface: it has a different symmetry group than a square: its symmetry group is the identity group.

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  • $\begingroup$ Can you elaborate please $\endgroup$ – user629838 Jan 1 at 17:25

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