# Point groups and symmetry: Adding and subtracting operations

• Jormungandr
In summary, the conversation discusses a series of short questions about symmetry for an inorganic class. The questions involve adding new elements to a group and finding all the different ways to combine them with the old group. This process can be non-intuitive and require a significant amount of work.
Jormungandr

## Homework Statement

I haven't been assigned these questions, but I'm trying to trudge through them to better understand symmetry. This is for my inorganic class.

It's just a series of short questions like:

C3 – S56 = ?
S4 + i = ?
C3 + i = ?

Stuff like this. And just looking at the elements in each group doesn't really help, because it's not just the given operation that's added, but sometimes several operations get added. I know the answer to each of these, but it's not really intuitive and I could use some help trying to think through them. Help is appreciated. Thanks!

well yes, you're right. when an extra element is added to the group, they really mean that the new group is all the elements that can be generated, by using the new element with old elements in the group. For example, in the case "C3 + i" to find out what the new group is, you need to find all the different ways to combine the new element with the old group. It's not too difficult in this specific case. But you're right that in general, it can be non-intuitive and require quite a lot of work.

## 1. What is a point group and why is it important in science?

A point group is a set of symmetry operations that describe the arrangement of atoms or molecules in a crystal or other structure. It is important in science because it helps us understand the physical and chemical properties of materials and molecules, and can also aid in predicting their behavior.

## 2. What are the different types of symmetry operations in a point group?

The different types of symmetry operations in a point group include rotation, reflection, inversion, and improper rotation. Rotation involves rotating the molecule or crystal around an axis, reflection involves reflecting the molecule or crystal across a plane, inversion involves swapping the positions of atoms or molecules, and improper rotation involves a combination of rotation and reflection.

## 3. How do you add symmetry operations in a point group?

To add symmetry operations in a point group, you simply apply each operation sequentially. For example, if you have a rotation operation followed by a reflection operation, you would first rotate the molecule or crystal, and then reflect it across a plane. The resulting symmetry operation would be a rotation-reflection.

## 4. Can symmetry operations be subtracted in a point group?

Yes, symmetry operations can be subtracted in a point group. This is known as the subtraction principle, where the resulting symmetry operation is the inverse of the subtraction of the two operations. For example, if you have a rotation operation followed by an inversion operation, the resulting symmetry operation would be a rotation-inversion.

## 5. How is knowledge of point groups and symmetry useful in crystallography?

Point groups and symmetry are essential in crystallography because they help determine the crystal structure of a material or molecule. By analyzing the symmetry of a crystal, scientists can determine the arrangement of atoms and predict the properties of the material. This information is crucial in various fields such as materials science, mineralogy, and chemistry.

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