- 579 Downloads
In Section 9.1 we show that if G is arguesian with respect to a hyperplane H, then the homothety group H H o is actually the multiplicative group of a field K H o whose underlying set is H H o ∪α o where α o is the endomorphism of G having H as kernel (and hence as axis) and o as constant value. One can consider α o as a degenerate homothety. Since the abelian group T H operates simply transitively on G\H one obtains on G\H an addition such that the map ev o : T H →G\H, the evaluation at o, becomes an isomorphism of groups. The set K H o also operates on G\H and one shows that the addition in G\H induces an addition in K H o , characterized by the equality (λ + μ)(x) = λ(x) + μ(x) for every x ∈ G\H. With this addition and the composition as multiplication K H o becomes a field. Furthermore, the set G\H with its addition and the scalar multiplication defined by λ · x : = λ (x) becomes a vector space over K H o . We denote this vector space by V H o or shortly by V.
Unable to display preview. Download preview PDF.