Why do groups capture the essence of symmetry?

[kexue]

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

 

[tiny-tim]

hi kexue!

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

 

[kexue]

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?

 

[tiny-tim]

But what about non-geometric objects … ?

perhaps i misunderstood your original question …

what did you mean by "symmetry" ?

 

[kexue]

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.)

 

[tiny-tim]

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

(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" ?

 

[A. Neumaier]

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.

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).