The Piecewise Projective Group as Group of Projectivities

1996 ◽  
Vol 30 (1-2) ◽  
pp. 122-135 ◽  
Author(s):  
Burkard Polster
Keyword(s):  
Projective Group ◽  
Group Of Projectivities

scholarly journals THE INVARIANTS FIELD OF SOME FINITE PROJECTIVE LINEAR GROUP ACTIONS

2011 ◽  
Vol 85 (1) ◽  
pp. 19-25
Author(s):  
YIN CHEN
Keyword(s):  
Finite Field ◽  
Projective Space ◽  
Vector Space ◽  
Orthogonal Group ◽  
Homogeneous Polynomial ◽  
Classical Group ◽  
Projective Group ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Projective Linear Group

AbstractLet Fq be a finite field with q elements, V an n-dimensional vector space over Fq and 𝒱 the projective space associated to V. Let G≤GLn(Fq) be a classical group and PG be the corresponding projective group. In this note we prove that if Fq (V )G is purely transcendental over Fq with homogeneous polynomial generators, then Fq (𝒱)PG is also purely transcendental over Fq. We compute explicitly the generators of Fq (𝒱)PG when G is the symplectic, unitary or orthogonal group.

scholarly journals Projective properties of Lorentzian surfaces

2019 ◽  
pp. 1-13
Author(s):  
Pierre Mounoud
Keyword(s):  
Infinite Order ◽  
Isometry Group ◽  
Projective Transformation ◽  
Compact Surface ◽  
Projective Group ◽  
Flat Torus

We investigate projective properties of Lorentzian surfaces. In particular, we prove that if T is a non-flat torus, then the index of its isometry group in its projective group is at most two. We also prove that any topologically finite non-compact surface can be endowed with a metric having a non-isometric projective transformation of infinite order.

On Subsemigroups of the Projective Group on the Line

1968 ◽  
Vol 20 ◽  
pp. 1001-1011 ◽  
Author(s):  
Franklin Lowenthal
Keyword(s):  
Projective Group ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Connected Component ◽  
Necessary And Sufficient ◽  
Infinitesimal Transformations

The subsemigroups of the projective group on the line that are described in this paper are those that can be generated by a pair of infinitesimal transformations. One denotes by G the connected component of the identity of this group; Theorem 1 gives a necessary and sufficient condition for a pair of infinitesimal transformations to generate a subsemigroup which is equal to G (and hence is actually a group). This condition is reformulated in a geometric manner in Theorem 1*.

Study of Certain Similitudes

1962 ◽  
Vol 14 ◽  
pp. 60-68 ◽  
Author(s):  
Maria J. Wonenburger
Keyword(s):  
Quadratic Form ◽  
Orthogonal Group ◽  
Scalar Multiplication ◽  
Projective Group ◽  
Classical Groups

One important point in the determination of the automorphisms of the classical groups is the study of group-theoretic properties of the elements of order 2, that is, the involutions. The group of similitudes and the projective group of similitudes of a non-degenerate quadratic form Q are extensions of the orthogonal group and the projective orthogonal group, respectively. These extended groups may contain involutions which do not belong to the orthogonal group or the projective orthogonal group. To study the automorphisms of such groups, group-theoretic properties of some of these new involutions should be established.The aim of this paper is to give some properties of a similitude T of ratio ρ whose square T2 is equal to the scalar multiplication by — p (it is always assumed that we are dealing with a field of characteristic ≠2). If ρ = — 1, T is an involution in the group of similitudes and for any ρ the coset of T in the projective group of similitudes is an involution.

On SO(3) as the Projective Group of the Space SO(3)

2020 ◽  
Vol 251 (4) ◽  
pp. 433-443
Author(s):  
A. A. Akopyan ◽  
A. V. Levichev
Keyword(s):  
Projective Group

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].

Lie subgroups of the symmetry group of the equations describing a nonstationary and isentropic flow: Invariant and partially invariant solutions

1994 ◽  
Vol 72 (7-8) ◽  
pp. 362-374 ◽  
Author(s):  
A. M. Grundland ◽  
L. Lalague
Keyword(s):  
Symmetry Group ◽  
Dimensional Space ◽  
Projective Group ◽  
Invariant Solutions ◽  
Partially Invariant Solutions ◽  
Isentropic Flow ◽  
Dimensional Space Time ◽  
Group A ◽  
Special Case

We study the symmetries of the equations describing a nonstationary and isentropic flow for an ideal and compressible fluid in four-dimensional space-time. We prove that this system of equations is invariant under the Galilean-similitude group. In the special case of the adiabatic exponent γ = 5/3, corresponding to a diatomic gas, the symmetry group of this system is larger. It is invariant under the Galilean-projective group. A representatives list of subalgebras of Galilean similitude and Galilean-projective Lie algebras, obtained by the method of classification by conjugacy classes under the action of their respective Lie groups, is presented. The results are given in a normalized list and summarized in tables. Examples of invariant and nonreducible partially invariant solutions, obtained from this classification, is constructed. The final part of this work contains an analysis of this classification in connection with a further classification of the symmetry algebras for the Euler and magnetohydrodynamics equations.

The Character Tables for SL(3, q), SU(3, q2), PSL(3, q), PSU(3, q2)

1973 ◽  
Vol 25 (3) ◽  
pp. 486-494 ◽  
Author(s):  
William A. Simpson ◽  
J. Sutherland Frame
Keyword(s):  
Character Table ◽  
Projective Group ◽  
Unitary Matrices ◽  
Character Tables

In this paper the character table of GL(3, q) (U(3, q2)), the group of all nonsingular n × n (unitary) matrices over GF(q) (GF(q2)), is used to obtain the character tables for the related subgroups SL(3, q), PSL(3, q) (SU(3, q2), PSU(3, q2)), the corresponding groups of matrices of determinant unity and the projective group respectively. There are very few abstract character tables which hold for entire families of groups. Such tables are of much greater value than tables for specific groups because, among other things, they enable one to discern various patterns common to the whole family.

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