For the present, this notation will be used although it is usually reserved for sets with a finite number of
elements. Later in this section, differentiable manifolds are discussed, for which is the symbol for
the group that is analogous to . As in the case of , invariance under holds in
the absence of any other geometrical structures on . A study of the subgroups of and their
relation to each other is thus equivalent to a study of all possible geometries on and their relation to each
other.