# Validity of Direct Product Structure of Symmetry Group

• Newtime
In summary: If we take two different reflections, we see that they commute. If we take an involution and the ##3-##cycle, we see that they also commute. So it is only left to show, that the involutions do not commute.In summary, the group of symmetries of a regular octahedron can be represented by a direct product of Z2, Z2, Z2, and S3. This is because the vertices can be grouped into three sets and the second factor is obtained by further permutation. When an element of the direct product is applied to a vertex, it is taken to "coordinates" similar to 3D coordinates. The group structure is determined by checking the commutativity of transformations and
Newtime
In short, if we consider the group of symmetries of a regular octahedron, we see (or at least, the author of "Groups, Graphs and Trees" saw...) that the group is isomoprhic to Z2$$\otimes$$Z2$$\otimes$$Z2$$\otimes$$S3 - particularly since if we break up the vertices into 3 groups of front-back, top-bottom and left-right we get the first three factors and the second factor is obtained by further permutation. But my question is how does an element of a direct product act on an element of a graph? If we take a vertex v in the graph, since all symmetries commute here, if we apply a symmetry h in the direct product to v, we are thus applying 4 symmetries each in one of the factors of the direct product. So is the vertex taken to "coordinates" of whatever the direct product indicates similar to the way we consider 3 dimensional coordinates as in if we have the coordinates (1,2,3) we could move 2 in the y direction then 1 in the x and 3 in the z or 3 in the z direction THEN 2 in the y then 1 in the x etc. and thus the same process for the location of the vertex v after h is applied. Thanks in advance for your help - I know I probably rambled a bit...

A direct product is written with ##\times## or ##oplus## for additionally written groups. ##\otimes## notes the tensor product. Anyway. You number the vertices of the octahedron and observe how group elements permute them. The group elements are defined as the geometric transformations which turn the octahedron onto itself, e.g. an upside down transformation. Then we have several reflection planes etc. The structure is defined by the way they form another transformation by consecutive application of two. E.g. upside down applied twice is the identity. It is a candidate for one copy of ##\mathbb{Z}_2##. In general it is not so easy to determine the group structure by its multiplication table. However, geometric figures are relatively simple. Let us assume all such transformations can be written as a product of three different involutions and a ##3-##cycle: ##T=I_1I_2I_3C##.
In case of a direct product, we get the structure by ##TT'=(I_1I_2I_3C)(I'_1I'_2I'_3C')=I_1I'_1I_2I'_2I_3I'_3CC'##. So we have to check, whether this is true for all the transformations.

## 1. What is the direct product structure of a symmetry group?

The direct product structure of a symmetry group refers to the way in which the group is composed of smaller subgroups. It is a way of combining these subgroups to form a larger group, allowing for a more efficient and organized way of representing the symmetries of an object or system.

## 2. How is the validity of the direct product structure determined?

The validity of the direct product structure is determined by analyzing the symmetry operations of the subgroups and how they combine to form the larger group. This involves examining the relationships between the subgroups and ensuring that they follow the rules of group theory.

## 3. What are the benefits of using the direct product structure in symmetry groups?

The direct product structure allows for a more organized and systematic way of representing symmetries, making it easier to understand and analyze the symmetries of an object or system. It also allows for the identification of patterns and relationships between different symmetries.

## 4. Are there any limitations to the direct product structure of symmetry groups?

While the direct product structure is a useful tool in understanding symmetries, it may not always accurately represent the symmetries of a complex system. In some cases, the direct product structure may oversimplify the symmetries or not fully capture all of the symmetries present.

## 5. How is the direct product structure used in practical applications?

The direct product structure is used in a variety of practical applications, particularly in the fields of physics and chemistry. It is used to analyze the symmetries of molecules and crystals, which can provide valuable information about their properties and behavior. It is also used in the study of particle physics to understand the symmetries of subatomic particles.

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