Collineation groups of the intersection of two classical unitals

2001 ◽  
Vol 9 (6) ◽  
pp. 445-459 ◽  
Author(s):  
L. Giuzzi
Keyword(s):  
Collineation Groups ◽  
Classical Unitals

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).

Conjecture of D. R. Hughes Extended to Generalized André Planes

1970 ◽  
Vol 22 (3) ◽  
pp. 701-704
Author(s):  
M. L. Narayana Ra
Keyword(s):  
Original Problem ◽  
Translation Planes ◽  
Alternate Proof ◽  
Collineation Groups

In 1967 Foulser [1] defined a class of translation planes, called generalized André planes or λ-planes and discussed the associated autotopism collineation groups. While discussing these collineation groups he raised the following question:“Are there collineations of a λ plane which move the axes but do not interchange them?”.In this context, Foulser mentioned a conjecture of D. R. Hughes that among the André planes, only the Hall planes have collineations moving the axes without interchanging them. Wilke [4] answered Foulser's question partially by showing that the conjecture of Hughes is indeed correct. Recently, Foulser [2] has shown that possibly with a certain exception the Hall planes are the only generalized André planes which have collineations moving the axes without interchanging them. Our aim in this paper is to give an alternate proof, which is completely general, and is in the style of the original problem.

A Family of Unitals in the Hughes Plane

1990 ◽  
Vol 42 (6) ◽  
pp. 1067-1083 ◽  
Author(s):  
Barbu C. Kestenband
Keyword(s):  
Present Article ◽  
English Translation ◽  
Symmetric Matrices ◽  
Configurational Property ◽  
The Future ◽  
The Matrix ◽  
Classical Unitals ◽  
Hughes Plane

We construct a family of unitals in the Hughes plane. We prove that they are not isomorphic with the classical unitals, and in so doing we exhibit a configuration that exists in the latter, but not in the former. This new configurational property of the classical unitals might serve in the future again as an isomorphism test.A particular instance of our construction has appeared in [11]. But it only concerns itself with the case where the matrix involved is the identity, whereas the present article treats the general case of symmetric matrices over a suitable field. Furthermore, [11] does not answer the isomorphism question. It states that (the English translation is ours) “It remains to be seen whether the unitary designs constructed in this note are isomorphic or not with known designs”.

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