Groups & Symmetry: Exploring the Connection

In summary, groups describe symmetry by capturing the fundamental properties of symmetry, such as closure, associativity, and inverse elements. This applies not only to geometric objects, but also to more abstract objects in various spaces. The group laws reflect the idea of transforming elements of an object and having it look the same afterwards, which leads to a product of two symmetries being another symmetry. This concept of symmetry is then generalized to more abstract objects, similar to the way the concept of numbers is generalized from natural numbers to more complex ones.
  • #1
kexue
196
2
Why do groups descibe symmetry? Why does a set which has an identity and inverse element, is closed under an abstract multplication operation and whose member obey the association law, captures symmetry?

Why is that?

thanks
 
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  • #2
hi kexue! :wink:

the rotations on any geometric object are closed, obey the associative law, and have an inverse

so they always form a group …

and if the geometric object has a symmetry, that means that there's a geometric operation which if repeated an integral number of times gives you the geometric identity, so any element of a group which has a power equal to the identity of the group represents a geometric symmetry :smile:
 
  • #3


the rotations on any geometric object are closed, obey the associative law, and have an inverse

so they always form a group …

But what about non-geometric objects and other symmetries than rotations? You mean we have a symmetry, translate it into some geometry, then rotate and translate it members and observe that the group laws are observed, so that finally we conclude symmetry is described by groups?
 
  • #4
kexue said:
But what about non-geometric objects … ?

perhaps i misunderstood your original question …

what did you mean by "symmetry" ?
 
  • #5


My definition of symmetry, roughly and perhaps: you transform elements of an object and the object looks the same afterwards. Why capture the group laws this procedure?

Or a better question: why do groups define symmetry? Why do we have symmetry when a set is group? (More then often is hard to see that we have symmetry in the sense I described above like transformaton of some point in space.)
 
  • #6
kexue said:
My definition of symmetry, roughly and perhaps: you transform elements of an object and the object looks the same afterwards.

well, then, we are talking about geometric objects, aren't we?
… why do groups define symmetry? Why do we have symmetry when a set is group?

we like to think in geometric terms, so when we define an abstract group, we like to find a geometric object (not necessarily in real space!) whose symmetries are represented by elements of the group :smile:
(More then often is hard to see that we have symmetry in the sense I described above like transformaton of some point in space.)

what do you mean by "transformaton of some point in space" ? :confused:
 
  • #7


kexue said:
My definition of symmetry, roughly and perhaps: you transform elements of an object and the object looks the same afterwards. Why capture the group laws this procedure?

Because ''looking the same'' is made precise by saying that you can go back (form the inverse symmetry). it also implies that you can apply the same or another symmetry to the same-looking thing and get another same-looking thing, so that the product of two
symmetries is another symmetry. Therefore, the symmetries of an object always form a group.

kexue said:
Or a better question: why do groups define symmetry? Why do we have symmetry when a set is group? (More then often is hard to see that we have symmetry in the sense I described above like transformaton of some point in space.)

Lots of groups describe symmetries in 2 or 3 space dimensions, where one can identify it with the usual notion of symmetry. From there, one simply generalizes the notion of symmetry to more abstract objects in more abstract spaces, in a similar way as the notion of a number is generalized from the ''natural'' numbers 1,2,3,... to more and more unnatural numbers such as 0, -1 2/3, sqrt(2), 1+sqrt(-1).
 

1. What is the concept of symmetry in relation to groups?

Symmetry refers to the balance and proportion in an object or system. In the context of groups, symmetry refers to the way in which elements within a group can be transformed or manipulated while still maintaining the same overall structure or pattern.

2. What are some examples of groups with symmetrical properties?

Groups with symmetrical properties can be found in many different areas of mathematics and science, such as crystallography, geometry, and physics. Some common examples include the dihedral group, the cyclic group, and the permutation group.

3. How are groups and symmetry connected?

Groups and symmetry are closely connected because groups are often used to describe and analyze the symmetrical properties of different objects and systems. In fact, the study of symmetry can provide insights into the structure and behavior of groups.

4. What practical applications does the study of groups and symmetry have?

The study of groups and symmetry has many practical applications in areas such as chemistry, physics, and computer science. For example, understanding the symmetrical properties of molecules can help chemists predict their reactivity, and symmetrical patterns are often used in coding and cryptography.

5. Are there any real-life examples of groups and symmetry?

Yes, there are many real-life examples of groups and symmetry, including the symmetrical patterns found in nature, such as snowflakes, flowers, and shells. Symmetry is also seen in man-made structures, such as buildings and bridges, which often have symmetrical designs for practical and aesthetic purposes.

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