Minkowski-Ebenen mit Spiegelungen

Dienst, Karl

doi:10.1007/bf01538030

Cited by 12 publications

(3 citation statements)

References 7 publications

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“…This doesn't arise confusion and agrees with the usually used notation (cf. [5], [6], [7]). From the definitions we remark that charM = 2 exactly when for some points p, q with p / ∈ [q] there exists a double homothety with centers p, q (cf.…”

Section: Proof By Theorem 22 There Existsmentioning

confidence: 99%

The symmetry axiom in Minkowski planes

Kosiorek

Matraś

2006

Abh.Math.Semin.Univ.Hambg.

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Section: Proof By Theorem 22 There Existsmentioning

confidence: 99%

The symmetry axiom in Minkowski planes

Kosiorek

Matraś

2006

Abh.Math.Semin.Univ.Hambg.

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“…The second part of this result does not appear explicitly in [4], but the author uses a theorem of Pickert which implies the entire property 3.1. Moreover, if M has finite order q, then q is a prime power and for i= 1, 2, T~, as a permutation group acting on an arbitrary 5r is isomorphic to PSL(2, q) in its usual representation over a projective line.…”

Section: Translationsmentioning

confidence: 99%

“…In other words, M is a well-known plane: it can be described by means of coordinates over a Tits nearfield in a way very similar to 1.3 (see Percsy [14] or [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]). …”

Section: 8mentioning

confidence: 99%

Finite Minkowski planes in which every circle-symmetry is an automorphism

Percsy

1981

Geom Dedicata

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NICOLAS PERCSY FINITE MINKOWSKI PLANES IN WHICH EVERY CIRCLE-SYMMETRY IS AN AUTOMORPHISM If a Minkowski plane is finite--i.e, the set of points is finite--all its lines and all its circles have the same number of points q + 1, where q ~> 2 is called the order of the plane. Actually, infinite planes, the 'tangency axiom' (T) is a consequence of the preceding conditions (Heise and Karzel I-6], Percsy [133). Two Minkowski planes M and M' are called isomorphic if there exists an isomorphism from M onto M', i.e. a one-to-one mapping preserving lines and circles. An isomorphism from M on M is called an automorphism of M. Examples of Minkowski PlanesA classical example of Minkowski plane (the ' Miquelian' plane) is given by the geometry of a ruled quadric Q in a three-dimensional projective space: M is the set of points of Q, ~x and ~2 are the two families of lines contained in Q, and the circles are the non-degenerate plane sections of Q.

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Invertible Sharplyn-Transitive Sets

Polster

1998

Journal of Combinatorial Theory, Series A

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We prove various results about sharply n-transitive sets of homeomorphisms of`n ice'' topological spaces like the real line and the circle. Our main results concern sharply 3-transitive sets G of homeomorphisms of the circle to itself. If G=G &1 , then G contains a real hyperbolic part (a set of involutions of the circle having special properties). We show that every real hyperbolic part is the hyperbolic part of a real abstract oval in the sense of Buekenhout and that every real abstract oval arises from a topological oval in a flat projective plane. This establishes a new relationship between flat Minkowski planes that admit automorphisms which are circle-symmetries and flat projective planes containing topological ovals. We also consider sharply n-transitive sets of permutations acting on finite sets. We find that our results about flat geometries and sharply n-transitive sets of homeomorphisms have counterparts in the finite case.

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Minkowski-Ebenen mit Spiegelungen

Cited by 12 publications

References 7 publications

The symmetry axiom in Minkowski planes

The symmetry axiom in Minkowski planes

Finite Minkowski planes in which every circle-symmetry is an automorphism

Invertible Sharplyn-Transitive Sets

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