A translation plane of order 72 with a very small translation complement

1993 ◽  
Vol 9 (2-4) ◽  
pp. 255-263 ◽  
Author(s):  
M. L. Narayana Rao ◽  
K. Kuppu Swamy Rao ◽  
Vinod Joshi
Keyword(s):  
Translation Plane ◽  
Translation Complement

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 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.

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 A characterization of the desarguesian planes of orderq2bySL(2,q)

1983 ◽  
Vol 6 (3) ◽  
pp. 605-608 ◽  
Author(s):  
D. A. Foulser ◽  
N. L. Johnson ◽  
T. G. Ostrom
Keyword(s):  
Translation Plane ◽  
Translation Complement ◽  
Desarguesian Planes

The main result is that if the translation complement of a translation plane of orderq2contains a group isomorphic toSL(2,q)and if the subgroups of orderqare elations (shears), then the plane is Desarguesian. This generalizes earlier work of Walker, who assumed that the kernel of the plane containedGF(q).

On a conjecture of Hughes

1967 ◽  
Vol 63 (3) ◽  
pp. 647-652 ◽  
Author(s):  
Judita Cofman
Keyword(s):  
Projective Plane ◽  
Permutation Group ◽  
Translation Plane ◽  
Collineation Group ◽  
Affine Plane ◽  
Finite Projective Plane ◽  
Transitive Permutation Group ◽  
Condition C

D. R. Hughes stated the following conjecture: If π is a finite projective plane satisfying the condition: (C)π contains a collineation group δ inducing a doubly transitive permutation group δ* on the points of a line g, fixed under δ, then the corresponding affine plane πg is a translation plane.

scholarly journals On spreads in PG(3,2s) that admit projective groups of order 2s

1985 ◽  
Vol 28 (3) ◽  
pp. 355-360
Author(s):  
V. Jha ◽  
N. L. Johnson
Keyword(s):  
Translation Complement ◽  
Group Of Projectivities

Let Γ be a spread in = PG(3, q); thus Γ consists of a set of q2 +1 mutually skew lines that partition the points of . Also let Λ be the group of projectivities of that leave Γ invariant: so Λ is the “linear translation complement” of Γ, modulo the kern homologies. Recently, inspired by a theorem of Bartalone [1], a number ofresults have been obtained, in an attempt to describe (Γ, Λ) when q2 divides |Λ|. A good example of such a result is the following theorem of Biliotti and Menichetti [3], which ultimately depends on Ganley's characterization of likeable functions of even characteristic [5].

scholarly journals A flag transitive plane of order49and its translation complement

1991 ◽  
Vol 14 (2) ◽  
pp. 339-344
Author(s):  
M. L. Narayana Rao ◽  
K. Satyanarayana ◽  
K. M. Arjuna Rao
Keyword(s):  
Solvable Group ◽  
Transitive Group ◽  
Translation Complement ◽  
Transitive Plane

The translation complement of the flag transitive plane of order49[Proc. Amer. Math. Soc. 32 (1972), 256-262] constructed by Rao is computed. It is shown that the flag transitive group itself is the translation complement and it is a solvable group of order600.

scholarly journals A translation plane of order 112

1983 ◽  
Vol 35 (3) ◽  
pp. 289-300 ◽  
Author(s):  
Mauro Capursi
Keyword(s):  
Translation Plane
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