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 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 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.

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.

Two-groups on finite translation planes

1986 ◽  
Vol 47 (6) ◽  
pp. 568-572 ◽  
Author(s):  
T. G. Ostrom
Keyword(s):  
Translation Planes ◽  
Finite Translation

scholarly journals Line-rank 3 affine planes

1982 ◽  
Vol 25 (3) ◽  
pp. 397-403
Author(s):  
Michael J. Kallaher ◽  
Graham Kelly
Keyword(s):  
Permutation Group ◽  
Collineation Group ◽  
Classical Result ◽  
Translation Planes ◽  
Affine Planes ◽  
Finite Affine ◽  
Dimension 2

We consider finite affine planes having a collineation group acting as a rank 3 permutation group on the affine lines. By a classical result of A. Wagner, such affine planes are translation planes. We show that if, in addition, the plane has odd dimension or dimension 2 over its kernel, then the plane is Desarguesian.

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