scholarly journals Translation planes of dimension two and characteristic two

1983 ◽  
Vol 6 (1) ◽  
pp. 41-58 ◽  
Author(s):  
N. L. Johnson ◽  
T. G. Ostrom
Keyword(s):  
Translation Planes ◽  
Translation Complement ◽  
Characteristic Two

This article discusses translation planes of dimension two and characteristic two. LetGbe a subgroup of the linear translation complement of such a planeπ. The nature ofGand its possible action onπare investigated. This continues previous work of the authors. It is shown that no new groups occur.

scholarly journals On a C–plane of order 25

1984 ◽  
Vol 30 (1) ◽  
pp. 27-36 ◽  
Author(s):  
M.L. Narayana Rao ◽  
K. Satyanarayana
Keyword(s):  
Interesting Property ◽  
Translation Planes ◽  
Translation Complement ◽  
Ideal Points ◽  
The Ideal

Rao, Rodabaugh, Wilke and Zemmer [J. Combin. Theory Ser. A. 11 (1971), 72–92] constructed a number of new VW systems called C-systems from the exceptional near–fields and established that they coordinatize translation planes not isomorphic to generalized André planes. In this paper the translation complement of the plane coordinatized by the C-system I–1 has been found. This plane has the interesting property that its translation complement divides the ideal points into two orbits of lengths 10 and 16. Further, the translation complement contains a subgroup isomorphic to SL(2,5) and therefore one of the exceptional Walker's planes of order 25 [H. Luneberg, Translation Planes, Springer-Verlag (1980), pp.235–244] is indeed the C–plane corresponding to the C–system I–1, which was discovered in 1969.

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.

scholarly journals A class of translation planes of square order

1984 ◽  
Vol 30 (1) ◽  
pp. 59-66 ◽  
Author(s):  
M.L. Narayana Rao ◽  
K. Satyanarayana ◽  
G. Vithal Rao
Keyword(s):  
Natural Number ◽  
Ideal Point ◽  
Salient Feature ◽  
Translation Planes ◽  
Translation Complement ◽  
Ideal Points

A class of translation planes of order p2r, where r is an odd natural number and p is a prime, p ≥ 7, p ≢ ± (mod 10) is constructed. A salient feature shared by all these planes is that one ideal point is fixed by the translation complement and the remaining ideal points are divided into at least two orbits, one of which is of length pr.

Translation Planes of Dimension Two with Odd Characteristic

1980 ◽  
Vol 32 (5) ◽  
pp. 1114-1125 ◽  
Author(s):  
T. G. Ostrom
Keyword(s):  
Vector Space ◽  
Vector Spaces ◽  
Translation Planes ◽  
Vector Form ◽  
Linear Transformations ◽  
Independent Vector ◽  
Translation Complement ◽  
Three Space ◽  
Zero Vector ◽  
Mutually Independent

A translation plane of dimension d over its kernel K = GF(q) can be represented by a vector space of dimension 2d over K. The lines through the zero vector form a “spread”; i.e., a class of mutually independent vector spaces of dimension d which cover the vector space.The case where d = 2 has aroused the most interest. The more exotic translation planes tend to be of dimension two; a spread in this case can be interpreted as a class of mutually skew lines in projective three-space.The stabilizer of the zero vector in the group of collineations is a group of semi-linear transformations and is called the translation complement. The subgroup consisting of linear transformations is the linear translation complement.

scholarly journals Translation complements of C-planes: (I)

1987 ◽  
Vol 36 (1) ◽  
pp. 99-111
Author(s):  
M. L. Narayana Rao ◽  
K. Kuppuswamy Rao ◽  
G. V. Subba Rao
Keyword(s):  
Translation Planes ◽  
Translation Complement ◽  
Finite Translation ◽  
New Class ◽  
Ideal Points

Narayana Rao, Rodabaugh, Wilke and Zemmer constructed a new class of finite translation planes from exceptional near-fields described by Dickson and Zassenhaus. These planes referred to as C-planes are not coordinatized by the generalized André systems. In this paper we compute the translation complement of the C-plane corresponding to the C-system III–1. It is found that the translation complement is of order 6912 and it divides the set of ideal points into two orbits of lengths 2 and 48.

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.

SOME CHARACTERIZATIONS OF DESARGUESIAN TRANSLATION PLANES

2006 ◽  
Vol 05 (01) ◽  
pp. 19-33
Author(s):  
DOUGLAS P. BROZOVIC ◽  
CHAT YIN HO
Keyword(s):  
Translation Planes ◽  
Translation Complement ◽  
Finite Translation ◽  
Finite Affine

In this note we consider finite translation planes with large translation complements. In particular, we characterize finite affine Desarguesian translation planes in two ways, according to the existence of subgroups in the translation complement that are divisible by relatively large integers, together with modest additional restrictions.

scholarly journals Translation planes of even order in which the dimension has only one odd factor

1980 ◽  
Vol 3 (4) ◽  
pp. 675-694 ◽  
Author(s):  
T. G. Ostrom
Keyword(s):  
Normal Subgroup ◽  
Translation Plane ◽  
Cyclic Subgroup ◽  
Prime Factor ◽  
Translation Planes ◽  
Translation Complement ◽  
Finite Translation ◽  
Finite Translation Plane ◽  
Irreducible Subgroup ◽  
Even Order

LetGbe an irreducible subgroup of the linear translation complement of a finite translation plane of orderqdwhereqis a power of2.GF(q)is in the kernel andd=2srwhereris an odd prime. A prime factor of|G|must divide(qd+1)∏i=1d(qi−1).One possibility (there are no known examples) is thatGhas a normal subgroupWwhich is aW-group for some primeW.The maximal normal subgroup0(G)satisfies one of the following:1. Cyclic. 2. Normal cyclic subgroup of indexrand the nonfixed-point-free elements in0(G)have orderr. 3.0(G)contains a groupWas above.

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