Proof of the ∞ Theorem

The ∞ Theorem
S(2) is the subgroup of I(2) consisting of elements which fix ∞.

We know that S(2) is generated by the elements of E(2),
and the scalings sk(z) = kz, for k > 0.
It follows that S(2) is generated by the elements of E(2)
and the extensions of the sk.
We have already observed that E(2) is a subgroup of I(2).
For k > 0, let k = r2.Then it is easy to see that iroi1(z) = kz,
so this element of I(2) is the extension of sk.
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 i-lines to i-lines. An i-line 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 :
    ir(z)=r2/z*, when z ≠ 0, ∞

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