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 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.

The Translation Planes of Dempwolff

1981 ◽  
Vol 33 (5) ◽  
pp. 1060-1073 ◽  
Author(s):  
N. L. Johnson
Keyword(s):  
Fixed Point ◽  
Vector Space ◽  
Translation Plane ◽  
Affine Plane ◽  
Translation Planes ◽  
Linear Transformations ◽  
Set Of Permutations ◽  
Transitive Set ◽  
Set Of Points

In [2], Dempwolff constructs three translation planes of order 16 using sharply 2-transitive sets of permutations in S16. That is, if acting on Λ is a sharply 2-transitive set of permutations then an affine plane of order n may be defined as follows: The set of points = {(x, y)|x, y ∊ Λ} and the lines = {(x, y)|y = xg for fixed }, {(x, y)|x = c}, {(x, y)|y = c} for c ∊ Λ.Let V be a vector space of dimension k over F ≅ GF(pr). A translation plane may be defined by finding a set M of pkr – 1 linear transformations such that xy–l is fixed point free on for all x ≠ y in M.Notice that if we allow V to act on itself then MV is a sharply 2-transitive set on V if and only if xy–1 is fixed point free on for all x ≠ y in M.

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 Translation planes of odd order and odd dimension

1979 ◽  
Vol 2 (2) ◽  
pp. 187-208 ◽  
Author(s):  
T. G. Ostrom
Keyword(s):  
Vector Space ◽  
Finite Groups ◽  
Translation Planes ◽  
Solvable Subgroup ◽  
Joint Paper ◽  
Desarguesian Planes ◽  
Odd Order ◽  
Collineation Groups ◽  
Zero Vector

The author considers one of the main problems in finite translation planes to be the identification of the abstract groups which can act as collineation groups and how those groups can act.The paper is concerned with the case where the plane is defined on a vector space of dimension2d overGF(q), whereqanddare odd. If the stabilizer of the zero vector is non-solvable, letG0be a minimal normal non-solvable subgroup. We suspect thatG0must be isomorphic to someSL(2,u)or homomorphic toA6orA7. Our main result is that this is the case whendis the product of distinct primes.The results depend heavily on the Gorenstein-Walter determination of finite groups having dihedral Sylow2-groups whendandqare both odd. The methods and results overlap those in a joint paper by Kallaher and the author which is to appear in Geometriae Dedicata. The only known example (besides Desarguesian planes) is Hering's plane of order27(i.e.,dandqare both equal to3) which admitsSL(2,13).

scholarly journals SOME INFINITE PERMUTATION GROUPS AND RELATED FINITE LINEAR GROUPS

2016 ◽  
Vol 102 (1) ◽  
pp. 136-149 ◽  
Author(s):  
PETER M. NEUMANN ◽  
CHERYL E. PRAEGER ◽  
SIMON M. SMITH
Keyword(s):  
Normal Subgroup ◽  
Vector Space ◽  
Finite Rank ◽  
Minimal Condition ◽  
Permutation Groups ◽  
Linear Groups ◽  
Normal Subgroups ◽  
Second Variation ◽  
Formula Group ◽  
Finite Linear

This article began as a study of the structure of infinite permutation groups $G$ in which point stabilisers are finite and all infinite normal subgroups are transitive. That led to two variations. One is the generalisation in which point stabilisers are merely assumed to satisfy min-n, the minimal condition on normal subgroups. The groups $G$ are then of two kinds. Either they have a maximal finite normal subgroup, modulo which they have either one or two minimal nontrivial normal subgroups, or they have a regular normal subgroup $M$ which is a divisible abelian $p$-group of finite rank. In the latter case the point stabilisers are finite and act irreducibly on a $p$-adic vector space associated with $M$. This leads to our second variation, which is a study of the finite linear groups that can arise.

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 Maximal normal subgroups of the integral linear group of countable degree

1976 ◽  
Vol 15 (3) ◽  
pp. 439-451 ◽  
Author(s):  
R.G. Burns ◽  
I.H. Farouqi
Keyword(s):  
Abelian Group ◽  
Automorphism Group ◽  
Normal Subgroup ◽  
Vector Space ◽  
Free Abelian Group ◽  
Normal Structure ◽  
Normal Subgroups ◽  
Finite Degree ◽  
Infinite Rank ◽  
Countably Infinite

This paper continues the second author's investigation of the normal structure of the automorphism group г of a free abelian group of countably infinite rank. It is shown firstly that, in contrast with the case of finite degree, for each prime p every linear transformation of the vector space of countably infinite dimension over Zp, the field of p elements, is induced by an element of г Since by a result of Alex Rosenberg GL(אo, Zp ) has a (unique) maximal normal subgroup, it then follows that г has maximal normal subgroups, one for each prime.

scholarly journals Normal subgroups in the group of column-finite infinite matrices

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Waldemar Hołubowski ◽  
Martyna Maciaszczyk ◽  
Sebastian Zurek
Keyword(s):  
Normal Subgroup ◽  
Vector Space ◽  
Division Ring ◽  
Classical Result ◽  
Normal Subgroups ◽  
Linear Transformations ◽  
Finite Dimensional ◽  
Infinite Cardinality ◽  
Positive Integers ◽  
Dimensional Range

Abstract The classical result, due to Jordan, Burnside, Dickson, says that every normal subgroup of GL ⁢ ( n , K ) \mathrm{GL}(n,K) , where 𝐾 is a field and n ≥ 3 n\geq 3 , which is not contained in the center contains SL ⁢ ( n , K ) \mathrm{SL}(n,K) . Rosenberg described the normal subgroups of GL ⁢ ( V ) \mathrm{GL}(V) , where 𝑉 is a vector space of any infinite cardinality dimension over a division ring. However, when he considers subgroups of the direct product of the center and the group of linear transformations 𝑔 such that g - id V g-\mathrm{id}_{V} has finite-dimensional range, the proof is incomplete. We fill this gap for countably dimensional 𝑉 giving description of the lattice of normal subgroups in the group of infinite column-finite matrices indexed by positive integers over any field.

On a Result of A. M. Macbeath on Normal Subgroups of a Fuchsian Group

1970 ◽  
Vol 13 (1) ◽  
pp. 15-16 ◽  
Author(s):  
W. Jonsson
Keyword(s):  
Fixed Point ◽  
Normal Subgroup ◽  
Exact Sequence ◽  
Fuchsian Group ◽  
Hyperbolic Plane ◽  
Orbit Space ◽  
Normal Subgroups ◽  
Lefschetz Fixed Point Formula ◽  
Image Position ◽  
Torsion Free

A. M. Macbeath, in November 1965, communicated the following theorem to me which he proved with the aid of the Lefschetz fixed point formula.Theorem. If Γ is a Fuchsian group and N a torsion free normal subgroup, then the rank of N/[Γ, N] is twice the genus of the orbit space D/Γ where D denotes the hyperbolic plane which Γ acts.This theorem will follow from a consideration of the exact sequence*

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