Group theory: point group of a cube

In summary, the point group of a cube is the same as its symmetry group, which includes rotations and reflections. This group has an order of 48 and can be represented by labeling the vertices. It is abstractly isomorphic to S_4 x C_2, which is the symmetric group on 4 letters direct product cyclic group of order 2. The group O_{h} is also known as the inversion symmetry group and can be written as O_{h} = S_{4} x C_{2}.
  • #1
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what is the point group of a cube?
 
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  • #2
it is the same as the ordinary symmetry group. (rotations and reflections). if the question was what's this group, then it is a group of order 48, you can get a presentation of it by labelling the vertices. It is abstractly isomorphic to S_4 x C_2, symmetric group on 4 letters direct product cyclic group of order 2.

To see the iso with S_4xC_2 imagine drawing in the diagonals of the box, there are 4 of them. the symmetries permute these 4 objects, the C_2 part is because there are reflections too.
 
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  • #3
If I remember my solid state course correctly, the group for a cube is [tex]O_{h} [/tex]. This takes into account the inversion symmetry. I believe it is written as [tex]O_{h} = S_{4} x C_{2} [/tex].

dt
 

1. What is a point group in group theory?

A point group in group theory is a mathematical concept used to describe the symmetries of an object. It is a set of all possible transformations (such as rotations, reflections, and inversions) that leave the object unchanged.

2. How many symmetry elements does a cube have?

A cube has 23 symmetry elements, which include 9 rotations, 6 reflections, and 8 improper rotations (rotations followed by a reflection).

3. What is the point group of a cube?

The point group of a cube is known as the Oh group, which stands for octahedral group. It is the group of all possible symmetries of a cube, including rotations, reflections, and improper rotations.

4. How is the point group of a cube represented?

The point group of a cube is typically represented using the Schoenflies notation or the Hermann-Mauguin notation. In Schoenflies notation, the point group of a cube is represented as Oh, while in Hermann-Mauguin notation, it is represented as m3m.

5. What are the applications of understanding the point group of a cube?

Understanding the point group of a cube has various applications in different fields such as crystallography, chemistry, and physics. It can be used to predict the physical and chemical properties of molecules and crystals, as well as to understand the symmetries of different objects in nature.

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