The symmetric group is, by definition, the group of all the permutations on points. These permutations are functions, as you pointed out.

These groups are called permutation groups because they are the groups which define the symmetry of a regular object with points. Such an object is best described in (I believe) -dimensions, where it can always be drawn without lines crossing over.

For examples, is the group of symmetries of a triangle, while is the group of symmetries of a tetrahedron.