The Crystallographic Point Groups
A crystallographic point group is a finite-dimensional group G of rotations and reflections each of which is a symmetry operation of a crystal sending it into itself with one fixed point through which pass the axes of rotations and planes of reflections. Thus G is a finite-dimensional subgroup of the Full Orthogonal Group O(3) consisting of 3rd order orthogonal matrices with determinants ±1. We note that the group D3 of symmetries of an equilateral triangle discussed in example 5 of section 1–2 is itself a crystallographic point group consisting only of rotations.
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- 4.H.J. Zassenhaurs, The Theory of Groups, Second Edition, Chelsea Pub. Co., New York, 1958.Google Scholar