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?the rotations on any geometric object are closed, obey the associative law, and have an inverse
so they always form a group …
well, then, we are talking about geometric objects, aren't we?My definition of symmetry, roughly and perhaps: you transform elements of an object and the object looks the same afterwards.
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… why do groups define symmetry? Why do we have symmetry when a set is group?
what do you mean by "transformaton of some point in space" ?(More then often is hard to see that we have symmetry in the sense I described above like transformaton of some point in space.)
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 twoMy 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?
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).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.)