Exploring Groups & Symmetry: Deciphering the Connection

In summary, the conversation discusses the notion of symmetry in relation to groups and algebraic concepts. There is a debate on the definition of symmetry and how it applies to different groups, such as SU(3) and E_6. The concept of quotient groups is also discussed, with examples given to understand its meaning. The conversation ends with a question about the isomorphism of SL_2(C)/(+-I) and the group of Mobius Transformations.
  • #1
Cincinnatus
389
0
I hear all the time people saying things like "groups are the algebraic equivalent to the notion of symmetry". Yet I don't think I've ever really understood what was meant by this. I'm familiar with groups of symmetries (like the dihedral groups), but in these cases, to me it just seems like we are using the language of groups to talk about the ordinary notion of symmetry.

It seems like people are saying all groups have "something to do with an algebraic notion of symmetry" What this might be though is unclear to me. For example, what does the group GL_2(R) have to do with symmetry?
 
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  • #2
Cincinnatus said:
It seems like people are saying all groups have "something to do with an algebraic notion of symmetry" What this might be though is unclear to me. For example, what does the group GL_2(R) have to do with symmetry?

If I tell you that the Lie group SU(3) (isomorphic to the set of 3x3 unitary matrices with determinant 1) is "a group which has something to do with an algebraic notion of some symmetry in nature", would you too have problem with that? What about say exceptional group [tex]E_6[/tex] or special orthogonal group [tex]SO(10)[/tex]??

How about the alternating group [tex]A_4[/tex]: the rotational symmetry group of a regular tetrahedron. In other words, it is "an algebraic notion" of the set of all inequivalent rotations of a tetrahedron. Would you find this more palatable than the above examples?

If so, I guess the reason you find that the [tex]A_4[/tex] situation is "clearer" because you can related to things that you are accustomed to. that doesn't automatically mean that [tex]SU(3)[/tex] is not a "symmetry" of some sort.

If we wish to study symmetry in a formal and consistent way, we need to define some rules, rules that are obviously motivated by things/symmetry that we are accustomed to (eg. rotation of a square, reflection of a triangle etc..). We've picked out the crucial features that make a symmetry a symmetry and call them axioms of group theory. But with these rules we can also study symmetries that are much harder to visualise (eg. rotations of 10-dimensional hypercube). And that's the power of mathematics and science can bring you.
 
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  • #3
"symmetry" has to be understood more broadly than just isometry.
 
  • #4
think about what symmetry MEAns. there are some motions, called symmetries, which can be composed, and which have inverses.

thats it. those ARE THE AXIOMS OF A GROUP.
 
  • #5
Thanks for your help.

I have another question, I'm also trying to get a more intuitive feel for what is meant by a quotient group.

I understand examples like R/(2piZ) (under addition), I can see how this is isomorphic to the circle group and to the group of complex numbers with modulus 1 (under multiplication).

When I try to think about other groups, for example: SL_2(C)/(+-I) I get quickly confused.

Is that group isomorphic to a direct product of some other group with itself? It seems to me like it should be... Clearly there is something about quotient groups that I am not understanding.
 
  • #6
Cincinnatus said:
Thanks for your help.When I try to think about other groups, for example: SL_2(C)/(+-I) I get quickly confused.

What is +-I ? Is this the purely imaginary numbers? If so, the answer is Z_2 = {0,1} under addition mod 2.
 
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  • #7
Sorry, I meant the set {+I,-I} where I is the identity matrix. SL_2(C)/(+-I) should be the group of Mobius Transformations.

I also don't see how you get Z_2 if (+-I) had meant the purely imaginary numbers...
 

1. What is the significance of exploring groups and symmetry?

The study of groups and symmetry is essential for understanding the underlying structure and patterns in nature, from the subatomic level to the macroscopic world. It allows us to identify and classify objects, as well as predict their behavior and properties.

2. How are groups and symmetry connected?

Groups and symmetry are closely related concepts. A group is a mathematical structure that describes the symmetry of an object, while symmetry refers to the invariance (or lack of change) of an object under certain transformations. In other words, groups mathematically represent the symmetries of an object.

3. What are some real-world examples of group and symmetry?

Groups and symmetry can be found in many different fields, including mathematics, physics, chemistry, biology, and art. For example, the symmetries of crystals can be described by groups, and the Laws of Motion in physics are invariant under certain transformations, demonstrating symmetry.

4. How does studying groups and symmetry benefit society?

Studying groups and symmetry has led to numerous breakthroughs and advancements in various industries, such as technology, medicine, and materials science. It also helps us understand and appreciate the beauty and complexity of the world around us.

5. What are some practical applications of group theory and symmetry?

Group theory and symmetry have practical applications in many areas, including cryptography, data compression, computer graphics, and pattern recognition. They are also used in the development of new materials, drugs, and technologies. Additionally, understanding group theory and symmetry can improve problem-solving skills and critical thinking abilities.

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