What does it mean ?
Consider a group composed by four elements ( a "discrete group" ).
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that I can write :
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The corresponding action is :
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Clearly it may reverse the x coordinate, the y coordinate, or the two.
Schematically :
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Now we may build the matrix :
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We can check such set of matrixes form a group.
Their determinant is :
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Check the inverse matrix is :
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whence :
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...SO(2) (called special orthogonal group) is a sub-group of O(2) (called orthogonal group) and we may form the matrixes a from the matrixes a through :
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By the way, many are redundant. For an example, if
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which means that changing ( x ---> - ; y ---> -y ) is equivalent to a rotation of p . See next figure.
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We know that matrixes :
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correspond to a simple rotation around the center of coordinates O.
What is the meaning of more general matrixes :
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From :
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we know that a corresponds to two combined operations :

- A symmetry with respect to axis OX , or OY , or both.
- A rotation a around the center of coordinates.

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On the figure is shown the succession of the two operations

It is clear that it is equivalent to a symmetry with respect to a straight line passing by O
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...We have enriched the "special orthogonal group " SO(2) which began the "orthogonal group" O(2). Then we discovers that this extended group contains mirror-symmetries : all the symmetries with respect to straight lines passing by the origin of coordinates O.
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