The author looks at affine planes from von Staudt's point of view, by investigating the consequences of regularity assumptions for the group Πa of affine projectivities. This group Πa, which consists of all products of parallel projections, is always doubly transitive; it is 2-regular only in Desarguesian affine planes, and it is 4-regular in free affine planes [A. Barlotti et al., Rend. Sem. Mat. Univ. Padova 60 (1978), 183--200; MR0555963 (81g:51005)]. Concerning 3-regularity, the author proves the following theorem: Let A be an affine plane, and assume that Πa is 3-regular. Then A is a translation plane, and if the kernel of A is not GF(2) (or if A is finite), then A is in fact Desarguesian.
On affine planes with 3-regular group of projectivities
FUNK, Martin
1985-01-01
Abstract
The author looks at affine planes from von Staudt's point of view, by investigating the consequences of regularity assumptions for the group Πa of affine projectivities. This group Πa, which consists of all products of parallel projections, is always doubly transitive; it is 2-regular only in Desarguesian affine planes, and it is 4-regular in free affine planes [A. Barlotti et al., Rend. Sem. Mat. Univ. Padova 60 (1978), 183--200; MR0555963 (81g:51005)]. Concerning 3-regularity, the author proves the following theorem: Let A be an affine plane, and assume that Πa is 3-regular. Then A is a translation plane, and if the kernel of A is not GF(2) (or if A is finite), then A is in fact Desarguesian.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
