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 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).
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- inversion in the circle |z|=r :
ir(z)=r2/z*, when z ≠ 0, ∞
ir(0)=∞,
ir(∞)=0.
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