Group PGL Research Articles

Liftings of Projective 2-Dimensional Galois Representations and Embedding Problems

In this paper we study liftings of 2-dimensional projective representations of the absolute Galois group of a field k. These liftings are related to solutions of certain Galois embedding problems. For representations with image isomorphic to one of the groups PSL(2, q) or PGL(2, q) with q ≡ ±3 (mod 8) or to one of the groups PSL(2, 7) or PSL(2, 9), and k of characteristic different from 2, we compute the obstruction to the existence of a lifting of index 2 (Theorems 3.6 and 3.7). To make this computation we consider the natural representations of these groups as permutation groups and then apply Serre′s formula on the Witt invariant of Tr(x2) given by Serre (Comment. Math. Helv. 59, 1984, 651-676). We also obtain from Crespo (C. R. Acad. Sci. Paris315, 1992, 625-628; C4 extensions of Sn as Galois groups, preprint, Barcelona, 1993; Central extensions of the alternating group as Galois groups, preprint, Barcelona, 1993; Galois realization of central extensions of the symmetric group with kernel a cyclic 2-group, preprint, Barcelona, 1993), a criterion for the existence of liftings of index 4 and 8 (Theorem 3.8). In the case k = Q, we exploit a result by Tate to obtain an easy computable criterion for the existence of a lifting with any given index (Proposition 4.1 and corollaries). The same kind of criterion can be used for other embedding problems over Q; in Theorem 4.4 we give a criterion for the existence of a solution to some embedding problems over the symmetric group. Section 1 contains preliminaries on projective representations and their liftings. In Section 2 we classify the cyclic central extensions of the groups PSL(2, q) and PGL(2, q), and establish the relation to liftings of the projective representations we are interested in. In Section 3 we find which degree 2 extensions of the symmetric and alternating group correspond to each degree 2 extension of PSL(2, q) and PGL(2, q) under the natural permutation representations, and then apply Serre′s formula. Section 4 is devoted to the case k = Q. In Section 5 some numerical examples are given.

Semisimple groups with the rewriting property Q5

Let n be an integer greater than 1, and let G be a group. A subset {x1, X2, ..., xn} of n elements of G is said to be rewritable if there are distinct permutations π and σ of {1,2, …, n} such that The group G is said to have the rewriting property Qn or to be n-rewritable, if every subset of n elements of G is rewritable. The main result of this paper shows that the only nontrivial semlsimple groups with the property Q5 are the alternating group A5, the symmetric group S5, the projective special linear group PSL(2, 7) and the projective general linear group PGL(2, 7)

On divisible designs and local algebras

AbstractWe study the action of the group PGL(m,A) on the projective space PG(m − 1,A) over a finite commutative local algebra A in order to construct a class of divisible designs, denoted by Dm(d,A), which is the classical one of 2‐designs (of points and of flats of fixed projective dimension) in the case where A is a field. We also study the constructed divisible designs with particular care for the case where d = m − 1. © 1995 John Wiley & Sons, Inc.

L functions in scattering on p-adic multiloop surfaces

The scattering processes on multiloop infinite graphs with one and the same valence p+1 of all vertices are studied herein. These graphs are discrete spaces of a constant negative curvature since they can be interpreted as quotients of the p-adic hyperbolic plane over free acting discrete subgroups of the projective group PGL(2,Qp). They are, in fact, identical to p-adic multiloop surfaces. Releasing from a graph its subgraph containing all loops, which is called the reduced graph Tred, we can get L functions corresponding to these finite closed graphs. For an infinite graph, the notion of spherical functions is introduced. These are the eigenfunctions of a discrete Laplace operator acting on the graph. s-wave scattering processes are considered and in this way they define the scattering amplitudes ci. In the discrete case, their number coincides with ‖Tred‖—the number of vertices of the reduced graph. Taking the product over all ci, we obtain the determinant of the scattering matrix which turns out to be presented as a ratio of two L functions: C∼L(α+)/L(α−). Here the L function is the Ihara–Selberg function depending only on the form of Tred. All the dependence on the initial value of p remains in arguments α± = t/2p ±√t2/4p2−1/p, t being the eigenvalue of the Laplacian. The Hashimoto–Bass theorem is used for expressing L function L(u) of any finite graph via the determinant of a local operator Δ(u) acting on this graph.

Isoperimetric normalization of planar curves

This paper presents an algorithm for transforming closed planar curves into a canonical form, independent of the viewpoint from which the original image of the contour was taken. The transformation that takes the contour to its canonical form is a member of the projective group PGL(2), chosen because PGL(2) contains all possible transformations of a plane curve under central projection onto another plane. The scheme relies on solving computationally an problem in which a transformation is sought which maximises the area of a curve given unit perimeter. In the case that the transformation is restricted to the affine subgroup there is a unique extremising transformation for any piecewise smooth closed curve. Uniqueness holds, almost always, even for curves that are not closed. In the full projective case, isoperimetric normalization is well-defined only for closed curves. We have found computational counterexamples for which there is more than one extremal transformation. Numerical algorithms are described and demonstrated both for the affine and the projective cases. Once a canonical curve is obtained, its isoperimetric area can be regarded as an invariant descriptor of shape. >

Character tables of certain association schemes coming from finite unitary and symplectic groups

In this paper we study the character tables of the association schemes obtain from the action of (i) finite unitary groups acting on the set of nonisotropic projective points, and (ii) finite symplectic groups acting on the set of nonisotropic projective lines. It is shown that these character tables are controlled by the character tables of corresponding smaller association schemes which are obtained from the action of the group PGL(2, q) on the cosets by cyclic subgroups Z q + 1 and Z q − 1 , respectively.

Halving PGL (2, 2 f ), f odd: A series of cryptocodes

A set S of permutations of k objects is μ-uniform, t-homogeneous if for every pair A, B of t-subsets of the ground set, there are exactly μ permutations in S mapping A onto B. Our main result (Theorem 1.2) is the construction of a (q − 1)-uniform, 2-homogeneous set of permutations of q + 1 objects contained in the projective group PGL(2, q), where q is a power of 2 with odd exponent. The main ingredient of the proof is a lemma concerning cubic equations in characteristic 2 (Lemma 2.6). The result is useful in the framework of theoretical secrecy and authentication. By a theorem of D.R. Stinson (Stinson 1990) one obtains families of cryptocodes which achieve perfect 2-fold secrecy and are 1-fold secure against spoofing (Corollary 1.3).

Sharply transitive subsets of P?L(2,F) and spreads covered by derivable partial spreads

It is well known that one can associate a spread and hence a translation plane with every flock of a Miquelian Minkowski plane. If the Miquelian Minkowski plane is described via the sharply 3-fold transitive action of the projective group PGL(2,F) on the projective line over the commutative fieldF then the flocks correspond to sharply transitive subsets of PGL(2,F). We show that under certain conditions one can also associate a spread with a sharply transitive subset of PΓL(2,F) where PΓL(2,F) acts on the projective line over the not necessarily commutative fieldF in the natural way.

Character tables of the association schemes of finite orthogonal groups acting on the nonisotropic points

We study the character tables of the association schemes obtained from the following actions of finite (simple) orthogonal groups: 1. (i) O 2 m +(2 n ) and O 2 m −(2 n ) acting on the set of nonisotropic points. 2. (ii) O 2 m +( p n ) and O 2 m −( p n ) ( p an odd prime) acting on each half of the set of nonisotropic points. 3. (iii) O 2 m + 1 ( p n ) ( p an odd prime) acting on the set of square-type and the set of non-square-type nonisotropic points. It is shown that the character tables of the association schemes in the above (i) and (ii) are controlled by the character table of the group PSL(2, q), while that the character tables of the association schemes in (iii) are controlled by the character tables of the association schemes obtained from the action of the group PGL(2, q) acting on the cosets by dihedral subgroups D 2( q − 1) and D 2( q + 1) , respectively.

A characterization of PGL(2, q), q odd

Following the lines of [10], we give a characterization of the group PGL(2, q), q odd, in terms of involutions.

On singular projective structures on Riemann surfaces

1 Let be a compact Riemann surface of genus g with function held J?(X) and let L=D’+AD+B (1.1) be a second order linear differential operator on X: D is a non-trivial derivation of A(X)/@ and A, B E A(X). We assume the singularities of L on to be all regular and denote by S a finite subset of containing all of them. Let X’ denote the punctured Riemann surface X S and let x0 E X’ be an ordinary point of L. Analytic continuation along closed paths of a ratio z of independent solutions of L at x0 yields a (class of) representa- tion(s) of the fundamental group x1(X’, x0) in the projective linear group PGL(2, C), which we call the projective monodromy representation pL of L. We may regard pL as a (conjugacy class of) group homomorphism(s) pr: x1(X’, x0) + PGL(2, a=) (1.2) or, via the natural identification, valid for any group G [S, Lemma 271 Hom(n,(X’, x,), G)/GrH’(X’, G), (1.3) as a cohomology class

PSL(2, q) as an image of the extended modular group with applications to group actions on surfaces

The modular group PSL(2, ℤ), which is isomorphic to a free product of a cyclicgroupof order 2 and a cyclic group of order 3, has many important homomorphic images. Inparticular, Macbeath [7] showed that PSL(2, q) is an image of the modular group if q ≠ 9. (Here, as usual, q is a prime power.) The extended modular group PGL(2, ℤ) contains PSL{2, ℤ) with index 2. It has a presentationthe subgroup PSL(2, ℤ) being generated by UV and VW.

Polyhedral groups and graph amalgamation products

A polyhedral group G is defined to be the orientation-preserving subgroup of a discrete reflection group acting on hyperbolic 3-space H 3, and having a fundamental polyhedron of finite volume. A special presentation for G is obtained from the geometry of the polyhedron. This gives G the structure of a graph amalgamation product, and which, in some cases, splits as a free product with amalgamation. The simplest examples of polyhedral groups are the so-called tetrahedral groups. Other examples are given amongst the the groups PGL(2, O m ), where O m is the ring of algebraic integers in the quadratic imaginary field Q( -m ) , m>0.

The mathematics of perfect shuffles

There are two ways to perfectly shuffle a deck of 2 n cards. Both methods cut the deck in half and interlace perfectly. The out shuffle O leaves the original top card on top. The in shuffle I leaves the original top card second from the top. Applications to the design of computer networks and card tricks are reviewed. The main result is the determination of the group 〈 I, O 〉 generated by the two shuffles, for all n. If 2 n is not a power of 2, and if 2 n ≠ 12,24, then 〈 I, O 〉 has index 1, 2, or 4 in the Weyl group B n (the group of all 2 n n! signed n × n permutation matrices). If 2 n = 2 k , then 〈 I, O 〉 is isomorphic to a semi-direct product of Z 2 k and Z k . When 2 n = 24, 〈 I, O 〉 is isomorphic to a semi-direct product of Z 2 11 and M 12, the Mathieu group of degree 12. When 2 n = 12, 〈 I, O 〉 is isomorphic to a semi-direct product of Z 2 6 and the group PGL(2,5) of all linear fractional transformations over GF(5).

Products of central collineations

An n by n matrix M over a (commutative) field F is said to be central if M − I has rank 1. We say that M is an involution if M 2= I; if M is also central we call M a simple involution. We will prove that any n-by- n matrix M satisfying det M=±1 is the product of n+2 or fewer simple involutions. This can be reduced to n+1 if F contains no roots of the equation x n =(−1) n other than ±1. Any ordered field is of this kind. Our main result is that if M is any n-by- n nonsingular nonscalar matrix and if x i ∈ F such that x 1⋯ x n =det M, then there exist central matrices M i such that M= M 1⋯ M n and x i =det M i for i=1,…, n. We will apply this result to the projective group PGL( n,F) and to the little projective group PSL( n,F).

Description of extraordinary representations of the group PGL(2, Q2)

Description of extraordinary representations of the group PGL(2, Q2)

Extensions of the Rationals with Galois Group Pgl(2,Zn )

Extensions of the Rationals with Galois Group Pgl(2,Z_n )

Spherical functions on the [formula omitted]-adic group PGL(2). I

Spherical functions on the [formula omitted]-adic group PGL(2). I

On Projective Invariance of Brownian Motion

Let E be a certain nuclear topological vector space contained in the Hilbert space (Z,2) on the real line and let the Gaussian probability measure be defined on the conjugate space E* of E. We consider such a subgroup of the rotation group of (L 2) that acts on E and contains the shift as a one-parameter subgroup. With a rather systematic way we define two more one-parameter subgroups which, together with the shift, constitute a subgroup isomorphic with the projective linear group PGL(2, R). It plays a role of the time change of the white noise. In this set up we formulate and prove the principle of projective invariance of the Brownian motion given by P. Levy.

On S2-Groups and Groups of Moebius Transformations

Let PGL(2, ƒ) denote the group of all Moebius transformationsover a field F. The object of this paper is to prove the following theorem.Theorem 1. G is an S2-group and the centre of G is trivial if and only if G is isomorphic to a group PGL(2, ƒ), char ƒ ≠ 2.This theorem was proved for finite groups in (1). The present paper extends the result to infinite groups and also improves the method of proof used in that paper. Many of the theorems given there were proved for infinite groups and are used here with appropriate references.