Lecture Notes 9: Multiplication on a Line
Given a line l in a Desarguesian affine plane and two points O and I on l, we define a multiplication of points A and C on l by:
Theorem: For any point A on l, AI = IA = A.
- Pick a point B not on l.
- Let m be the line through A parallel (or equal) to the line through I and B.
- Let D be the intersection of m and the line through O and B.
- Let k be the line through D parallel (or equal) to the line through B and C.
- Define AC to be the intersection of k and l.
Theorem: For any point A on l, AO = OA = O.
Theorem: If AO, there is a point (A-1) on l such that A(A-1) =(A-1)A = I.
Theorem: The definition of the point AC is independent of the choice of B.
Theorem: For any points A, C and E on l, (AC)E = A(CE).
The above theorems show that the operation of multiplication on the points of a line turns that set into a group (not necessarily abelian).