In group theory, a group $G$ can often be divided into classes. Two elements $U,V \in G$ belong to the same class if $\exists R\in G: RUR^{-1}=V$. This always strikes me as counter-intuitive. Apart from groups like symmetry groups, where one can assign "physical meaning" to the group elements, I lack understanding/a reason for why this distinction of classes is as powerful as it is. Why *would* an **abstract** group be divisible into classes? (Maybe a historical argument can help.)

Of course, I do not doubt that it is possible to do this when looking at concrete group tables and figuring out the classes manually. Is there an underlying reason for why dividing the group is most of the time possible with classes that do not only consist of one or all elements of the group?

~~Another thing is, why can an element only belong to exactly one class? The definition seems a bit lax; I sometimes wonder why there cannot be to different $R\_1,R\_2$ that sort some element into two different classes?~~
Because then that is actually the same class for both elements...

Edit
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This also comes from the fact that, when trying to devide a group into classes given only the group table and no "visual representation" (e.g. symmetry operations) for the group elements, how can at some point I know that I have found all elements of the class; that is, without calculating all possible combinations $RUR^{-1}$? "There might be some other combination of elements, that connects two classes..."