By Günter Pickert (auth.), Peter Plaumann, Karl Strambach (eds.)
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E. every point on a and every line through Care kept fixed); for CEa the elements of r(C,a) are called eZations. Then the Moufang planes can be characterized also by the transitivity of r(C,a) on 1 ,{C}, where 1 is a line ~a through C, for all incident pairs (C,a). Thus every perspectivity of b to c from C is induced by an elation with center C 33 PROJECTIVITIES IN PROJECTIVE PLANES and axis a = (bA c)C (see fig. 11). Therefore the stabilizer f l , P of 1,P in the group f. generated by all elations, induces nl • p • Now the group f(P) of all elations with center P is a ng.
KIST will find applications in circle geometries. § 1 BASIC ALGEBRAIC CONCEPTS Permutation sets. A set E of the symmetric group (E,I:) • For n EN SE together with a subset Z is called a permutation set a permutation set (E,L:) is called n-transi ti ve if for any two n-tuples (a/I ' a 2 , ••• , an) , ( b /1 ' b 2 ' • • • , b n) E En wi th I {a/I ' • • • , an} I = I {b /1 ' • • • , b n} I = n there is a O' EL: with o(a i ) =b i for i €{1,2, ... ,n} . (E,L:) n has the property (Pn) if only the identity fixes distinct points of n-transitive if ty (Pn)' E, (E,L:) and regular if (E,L:) is called sharply is n-transitive with the proper(E,L:) is sharply /I-transitive.
This development is most impressively reflected by the bibliography of BENZ's book [4J. Circle geometries, as the term is used in our investigations, were first considered inthe last century in the works of MOBIUS, LIE and LAGUERRE. Itwas not until the paper of B. L. van der WAERDEN and L. J. SMID that modern methods were introduced in the research of circle geometries. AIso PETKANTSCHIN investigated circle geometries in 1940. Both papers received no attention for quite a while due to difficult times.