definition of point group symmetry in chemistry
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geometry, a point group is a group of geometric symmetries (isometries) that keep at least one point fixed. Point groups can exist in a Euclidean space with any dimension, and every point group in dimension d is a subgroup of the orthogonal group O(d). Point groups can be realized as sets of orthogonal matrices M that transform point x into point y:
y = Mx
where the origin is the fixed point. Point-group elements can either be rotations (determinantof M = 1) or else reflections, or improper rotations (determinant of M = −1).
Discrete point groups in more than one dimension come in infinite families, but from the crystallographic restriction theorem and one of Bieberbach's theorems, each number of dimensions has only a finite number of point groups that are symmetric over some lattice or grid with that number. These are the crystallographic point groups.
y = Mx
where the origin is the fixed point. Point-group elements can either be rotations (determinantof M = 1) or else reflections, or improper rotations (determinant of M = −1).
Discrete point groups in more than one dimension come in infinite families, but from the crystallographic restriction theorem and one of Bieberbach's theorems, each number of dimensions has only a finite number of point groups that are symmetric over some lattice or grid with that number. These are the crystallographic point groups.
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