Ternary forms as invariants of Markoff forms and other SL2 (Z)-bundles

Consider the free group Γ = {A,B} generated by matrices A, B in SL₂(Z). We can construct a ternary form Φ(x,y,z) whose GL₃(Z) equivalence class is invariant, as it depends on Γ and not the choice of generators. If Γ is the commutator of SL₂(Z), then the generating matrices have fixed points corresponding to different fields and inequivalent Markoff forms, but they are all biuniquely determined by Φ = -z²+ y(2x+y+z) to within equivalence. When referred to transformations A, B of the upper half plane, this phenomenon is interpreted in terms of inequivalent homotopy elements which are primitive for the perforated torus.     

Galois number fields with small root discriminant

We pose the problem of identifying the set K(G,Ω) of Galois number fields with given Galois group G and root discriminant less than the Serre constant Ω≈44.7632. We definitively treat the cases G=A₄, A₅, A₆ and S₄, S₅, S₆, finding exactly 59, 78, 5 and 527, 192, 13 fields, respectively. We present other fields with Galois group SL₃(2), A₇, S₇, PGL₂(7), SL₂(8), ΣL₂(8), PGL₂(9), PΓL₂(9), PSL₂(11), and , and root discriminant less than Ω. We conjecture that for all but finitely many groups G, the set K(G,Ω) is empty.     

Quotients of the congruence kernels of SL2 over arithmetic dedekind domains

LetC=C(C, P, k) be the coordinate ring of the affine curve obtained by removing a closed pointP from a (suitable) projective curveC over afinite fieldk. Let SL₂ (C,q) be the principal congruence subgroup of SL₂(C) andU ₂(C,q) be the subgroup generated by the all unipotent matrices in SL₂(C,q), whereq is aC-ideal. In this paper we prove that, for all but finitely manyq, the quotient SL₂(C,q)/U ₂(C,q) is a free group of finite,unbounded rank. LetC(SL₂(A)) be the congruence kernel of SL₂(A), whereA is an arithmetic Dedekind domain with only finitely many units. (e.g.A=C or ℤ) and letG be any finitely generated group. From the above (and previous results) we deduce that the profinite completion ofG,Ĝ, is a homonorphic image ofC(SL₂(A)). This is related to previous results of Lubotzky and Mel'nikov.     

Isomorphy classes of finite order automorphisms of SL(2,k)

In this paper, we consider the order m k-automorphisms of SL(2,k). We first characterize the forms that order m k-automorphisms of SL(2,k) take and then we find simple conditions on matrices A and B, involving eigenvalues and the field that the entries of A and B lie in, that are equivalent to isomorphy between the order m k-automorphisms Inn_A and Inn_B. We examine the number of isomorphy classes and conclude with examples for selected fields.     

The Groups of Motions of Some Three-Dimensional Maximal Mobility Geometries

We find the groups of motions of eight three-dimensional maximal mobility geometries. These groups are actions of just three Lie groups SL₂(R)ΔN, SL₂(C) _R , and SL₂(R)?SL₂(R) on the space R³, where N is a normal abelian subgroup. We also find explicit expressions for these actions.     

Noncommutative differential geometry on the quantum SU(2), I: An algebraic viewpoint

We compute the cyclic homology of the coordinate ring A(SL_q(2)) of the quantum algebraic group SL _q (2). We observe a degeneration of the noncommutative de Rham complex. The results are also verified from the point of view of Connes' noncommutative differential geometry.     

On the second cohomology of an algebraic group and of its lie algebra in a positive characteristic

Necessary and sufficient isomorphism conditions for the second cohomology group of an algebraic group with an irreducible root system over an algebraically closed field of characteristic p ≥ 3h ? 3, where h stands for the Coxeter number, and the corresponding second cohomology group of its Lie algebra with coefficients in simple modules are obtained, and also some nontrivial examples of isomorphisms of the second cohomology groups of simple modules are found. In particular, it follows from the results obtained here that, among the simple algebraic groups SL₂(k), SL₃(k), SL₄(k), Sp₄(k), and G ₂, nontrivial isomorphisms of this kind exist for SL₄(k) and G ₂ only. For SL₄(k), there are two simple modules with nontrivial second cohomology and, for G ₂, there is one module of this kind. All nontrivial examples of second cohomology obtained here are one-dimensional.     

The construction of SL(2, 3)-polynomials

Octic polynomials over $\text{Z}$ with Galois group SL(2, 3) are constructed. This is done via suited quartic totally real polynomials with group A₄ over $\text{Q}$. A table of the cycle patterns of the imprimitive transitive permutation groups of degree 8 is included.     

On the First Galois Cohomology Group of the Algebraic Group SL1(D)

Let A be a central simple algebra over a field F. Denote the reduced norm of A over F by Nrd: A* → F* and its kernel by SL₁(A). For a field extension K of F, we study the first Galois Cohomology group H ¹(K,SL₁(A)) by two methods, valuation theory for division algebras and K-theory. We shall show that this group fails to be stable under purely transcendental extension and formal Laurent series.     

Weight reduction for mod ? Bianchi modular forms

Let K be an imaginary quadratic field with class number one and ? be a rational prime that splits in K. We prove that mod ?, a system of Hecke eigenvalues occurring in the first cohomology group of some congruence subgroup Γ of SL₂(OK) can be realized, up to twist, in the first cohomology with trivial coefficients after increasing the level of Γ by (?).     

On Non-Congruence Subgroups of the Analogue of the Modular Group in Characteristic p

Let k[t] be the polynomial ring over a finite field k. The group SL ₂(k[t]) is often referred to as the analogue, in characteristic p, of the classical modular group SL ₂( ), where is the ring of rational integers. It is well-known that the smallest index of a non-congruence subgroup of SL ₂( ) is 7. Here we compute this index for SL ₂(k[t]). (In all but 6 cases it turns out to be 1 + q, where q is the order of k.)     

THE BANACH-LIE GROUP OF LIE TRIPLE AUTOMORPHISMS OF AN H^＊-ALGEBRA^＊

We study the Banach-Lie group Ltaut(A) of Lie triple automorphisms of a complex associative H*-algebra A. Some consequences about its Lie algebra, the algebra of Lie triple derivations of A, Ltder(A), are obtained. For a topologically simple A, in the infinite-dimensional case we have Ltaut(A)0 = Aut(A) implying Ltder(A) = Der(A). In the finite-dimensional case Ltaut(A)0 is a direct product of Aut(A) and a certain subgroup of Lie derivations δ from A to its center, annihilating commutators.     

Periodic Groups Saturated with the Groups <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript>(<Emphasis Type="Italic">p</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>)

Given an indexing set I and a finite field K_α for each α ∈ I, let ? = {L₂(K_α) | α ∈ I} and \(\mathfrak{N} = \{ SL_2 (K_\alpha )|\alpha \in I\}\). We prove that each periodic group G saturated with groups in \(\Re (\mathfrak{N})\) is isomorphic to L₂(P) (respectively SL₂(P)) for a suitable locally finite field P.     

Finite Groups with Given Quantitative Non-Nilpotent Subgroups II

As an extension of Shi and Zhang's 2011 article [4 Shi , J. , Zhang , C. ( 2011 ). Finite groups with given quantitative non-nilpotent subgroups . Comm. Algebra 39 : 3346 – 3355 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]], we prove that any finite group having at most 23 non-normal non-nilpotent proper subgroups is solvable except for G ? A ₅ or SL(2, 5), and any finite group having at most three conjugacy classes of non-normal non-nilpotent proper subgroups is solvable except for G ? A ₅ or SL(2, 5).     

Finite classical groups and flag-transitive triplanes

Let D be a finite nontrivial triplane, i.e. a 2-(v,k,3) symmetric design, with a flag-transitive, point-primitive automorphism group G. If G is almost simple, with the simple socle X of G being a classical group, then D is either the unique (11, 6, 3)-triplane, with G=PSL₂(11) and G_α=A₅, or the unique (45, 12, 3)-triplane, with G=X:2=PSp₄(3):2≅PSU₄(2):2 and , where α is a point of D.     

A combinatorial result related to the consistency of New Foundations

We prove a combinatorial result for models of the 4-fragment of the Simple Theory of Types (TST), TST₄. The result says that if A=〈A₀,A₁,A₂,A₃〉 is a standard transitive and rich model of TST₄, then A satisfies the 〈0,0,n〉-property, for all n≥2. This property has arisen in the context of the consistency problem of the theory New Foundations (NF). The result is a weak form of the combinatorial condition (existence of ω-extendible coherent triples) that was shown in Tzouvaras (2007) [5] to be equivalent to the consistency of NF. Such weak versions were introduced in Tzouvaras (2009) [6] in order to relax the intractability of the original condition. The result strengthens one of the main theorems of Tzouvaras (2007) [5, Theorem 3.6] which is just equivalent to the 〈0,0,2〉-property.     

Synchronization of two low-dimensional Kerr chains

Synchronization of two chaotic low-dimensional chains (α₁, α₂, α₃) and (A₁, A₂, A₃) consisting of Kerr oscillators is studied. The synchronization has been achieved by the parallel coupling of α₁ with A₁, α₂ with A₂ and α₃ with A₃. We want to find whether and when the pairs (α₁, A₁), (α₂, A₂) and (α₃, A₃) synchronize non-simultaneously (three-time synchronism). The problem of synchronization is also studied for a number of couplings between the chains lower than the number of oscillators in a single chain. Both the ring and linear geometry of synchronization is investigated. The presented results suggest a possibility of multi-time synchronism in two coupled high-dimensional chains. It seems very promising for design of some devices for advanced signal processing.     

On quadruples of Griffiths points

Tabov (Math Mag 68:61–64, 1995) has proved the following theorem: if points A ₁, A ₂, A ₃, A ₄ are on a circle and a line l passes through the centre of the circle, then four Griffiths points G ₁, G ₂, G ₃, G ₄ corresponding to pairs (Δ _i ,l) are on a line (Δ _i denotes the triangle A _j A _k A _l , j,k,l ≠ i). In this paper we present a strong generalisation of the result of Tabov. An analogous property for four arbitrary points A ₁, A ₂, A ₃, A ₄, is proved, with the help of the computer program “Mathematica”.     

Slight improvements of the Singleton bound

The problem of determining A_q(n,d), the maximum cardinality of a q-ary code of length n with minimum distance at least d, is considered in some cases where corresponding MDS codes do not exist. Slight improvements of the Singleton bound are given, including A_q(q+2,q)?q³-3 if q is odd, A₅(7,5)?5³-4 and A₁₆(18,15)?18⁴-4.