Normal subgroups of collineation groups of finite translation planes

1974 ◽  
Vol 2 (4) ◽  
Author(s):  
T.G. Ostrom
Keyword(s):  
Translation Planes ◽  
Normal Subgroups ◽  
Finite Translation ◽  
Collineation Groups

scholarly journals Collineation groups of translation planes of small dimension

1981 ◽  
Vol 4 (4) ◽  
pp. 711-724 ◽  
Author(s):  
T. G. Ostrom
Keyword(s):  
Fixed Point ◽  
Normal Subgroup ◽  
Vector Space ◽  
Translation Plane ◽  
Small Dimension ◽  
Translation Planes ◽  
Normal Subgroups ◽  
Translation Complement ◽  
Collineation Groups

A subgroup of the linear translation complement of a translation plane is geometrically irreducible if it has no invariant lines or subplanes. A similar definition can be given for “geometrically primitive”. If a group is geometrically primitive and solvable then it is fixed point free or metacyclic or has a normal subgroup of orderw2a+bwherewadivides the dimension of the vector space. Similar conditions hold for solvable normal subgroups of geometrically primitive nonsolvable groups. When the dimension of the vector space is small there are restrictions on the group which might possibly be in the translation complement. We look at the situation for certain orders of the plane.

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.

scholarly journals On collineation groups of translation planes of orderq4

1986 ◽  
Vol 9 (3) ◽  
pp. 617-620
Author(s):  
V. Jha ◽  
N. L. Johnson
Keyword(s):  
Translation Plane ◽  
Translation Planes ◽  
Nonsolvable Group ◽  
Translation Complement ◽  
Affine Translation ◽  
Collineation Groups ◽  
Affine Translation Plane

LetPbe an affine translation plane of orderq4admitting a nonsolvable groupGin its translation complement. IfGfixes more thanq+1slopes, the structure ofGis determined. In particular, ifGis simple thenqis even andG=L2(2s)for some integersat least2.

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