Linear collineation groups preserving an arc in a Möbius plane

1999 ◽  
Vol 197-198 (1-3) ◽  
pp. 749-757
Author(s):  
A Sonnino
Keyword(s):  
Collineation Groups ◽  
Möbius Plane

A Characterization of Some Geometries of Chains

1974 ◽  
Vol 26 (02) ◽  
pp. 257-272 ◽  
Author(s):  
Yi Chen
Keyword(s):  
Projective Plane ◽  
Minkowski Plane ◽  
Close Relation ◽  
Laguerre Plane ◽  
Geometric Structures ◽  
Möbius Plane

The geometries considered here are the Möbius plane M() (W. Benz [1]), the Laguerre plane L() (W. Benz and H. Mäurer [7]) and the Minkowski plane A() (W. Benz [5], G. Kaerlein [18]) over a field . All of them are geometries of an algebra with identity over a field. The characterization of the projective plane over a field by the proposition of Pappus first gave a close relation between algebraic and geometric structures. B. L. v. d. Waedern and L. J. Smid [28] presented a further example by characterizing the Möbius and Laguerre plane with incidence axioms and the "complete" proposition of Miquel.

On Finite Groups with an Abelian Sylow Group

1962 ◽  
Vol 14 ◽  
pp. 436-450 ◽  
Author(s):  
Richard Brauer ◽  
Henry S. Leonard
Keyword(s):  
Finite Groups ◽  
Irreducible Characters ◽  
Collineation Groups ◽  
Sylow Group

We shall consider finite groups of order of g which satisfy the following condition:(*) There exists a prime p dividing g such that if P ≠ 1 is an element of p-Sylow group ofthen the centralizer(P) of P incoincides with the centralizer() of in.This assumption is satisfied for a number of important classes of groups. It also plays a role in discussing finite collineation groups in a given number of dimensions.Of course (*) implies that is abelian. It is possible to obtain rather detailed information about the irreducible characters of groups in this class (§ 4).

scholarly journals Affine planes with primitive collineation groups

1990 ◽  
Vol 128 (2) ◽  
pp. 366-383 ◽  
Author(s):  
Yutaka Hiramine
Keyword(s):  
Affine Planes ◽  
Collineation Groups

Collineation Groups of Generalized André Planes

1969 ◽  
Vol 21 ◽  
pp. 358-369 ◽  
Author(s):  
David A. Foulser
Keyword(s):  
Main Question ◽  
Translation Planes ◽  
Hall Plane ◽  
Collineation Groups

In a previous paper (5), I constructed a class of translation planes, called generalized André planes or λ-planes, and discussed the associated autotopism collineation groups. The main question unanswered in (5) is whether or not there exists a collineation η of a λ-plane Π which moves the two axes of Π but does not interchange them.The answer to this question is “no”, except if Π is a Hall plane (or possibly if the order n of Π is 34) (Corollary 2.8). This result makes it possible to determine the isomorphisms between λ-planes. More specifically, let Π and Π′ be two λ-planes of order n coordinatized by λ-systems Qand Q′, respectively. Then, except possibly if n = 34, Π and Π′ are isomorphic if and only if Q and Q′ are isotopic or anti-isotopic (Corollary 2.13). In particular, Π is an André plane if and only if Q is an André system (Corollary 2.14).

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