The ∞ Theorem
S(2) is the subgroup of I(2) consisting of elements which fix ∞.
Proof
We know that S(2) is generated by the elements of E(2), and the
scalings s_{k}(z) = kz, for k > 0.
It follows that S(2) is generated by the
elements of E(2) and the extensions
of the s_{k}.
We have already observed that E(2) is a subgroup of I(2).
For k > 0, let k = r^{2}.Then it is easy to see that
i_{r}oi_{1}(z) = kz,
so this element of I(2) is the extension of s_{k}.
Thus S(2) is a subgroup of I(2). By the way we extended
elements of S(2), each element of S(2) fixes ∞.
Suppose now that tεI(2) fixes ∞.
Now t maps ilines to ilines. An iline L is an extended line
if and only if ∞ ε L.
Thus t maps extended lines to extended lines.
If we restrict it to E, t maps lines to lines.
As t is inversive, it preserves angles.
Thus, t (restricted to E) is a similarity (see similarity page)
so
we have t ε S(2).

 inversion in the circle z=r :
i_{r}(z)=r^{2}/z*, when z ≠ 0, ∞
i_{r}(0)=∞,
i_{r}(∞)=0.
