How Do Groups Relate to Symmetry in Mathematics?

[Cincinnatus]

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?

 

[mjsd]

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 E_6 or special orthogonal group SO(10)??

How about the alternating group A_4: 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 A_4 situation is "clearer" because you can related to things that you are accustomed to. that doesn't automatically mean that SU(3) 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.

 

[mathwonk]

"symmetry" has to be understood more broadly than just isometry.

 

[mathwonk]

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.

 

[Cincinnatus]

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.

 

[Crosson]

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.

 

[Cincinnatus]

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...