Nontrivial Quotient Research Articles

Finite Groups Whose Maximal Subgroups are 2-Nilpotent or Normal

We describe the structure of those finite groups whose maximal subgroups are either 2-nilpotent or normal. Among other properties, we prove that if such a group G does not have any non-trivial quotient that is a 2-group, then G is solvable. Also, if G is a solvable group satisfying the above conditions, then the 2-length of G is less than or equal to 2. If, on the contrary, G is not solvable, then G has exactly one non-abelian principal factor and the unique simple group involved is one of the groups PSL2(p2a)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{PSL}_2(p^{2^a})$$\\end{document}, where p is an odd prime and a≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a\\ge 1$$\\end{document}, or p is a prime satisfying p≡±1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p\\equiv \\pm 1$$\\end{document}(mod8)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\ extrm{mod}~ 8)$$\\end{document} and a=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a=0$$\\end{document}.

FINITENESS OF CANONICAL QUOTIENTS OF DEHN QUANDLES OF SURFACES

Abstract The Dehn quandle of a closed orientable surface is the set of isotopy classes of nonseparating simple closed curves with a natural quandle structure arising from Dehn twists. In this paper, we consider the finiteness of some canonical quotients of these quandles. For a surface of positive genus, we give a precise description of the 2-quandle of its Dehn quandle. Further, with some exceptions for genus more than 2, we determine all values of n for which the n-quandle of its Dehn quandle is finite. The result can be thought of as the Dehn quandle analogue of a similar result of Hoste and Shanahan for link quandles [‘Links with finite n-quandles’, Algebr. Geom. Topol.17(5) (2017), 2807–2823]. We also compute the size of the smallest nontrivial quandle quotient of the Dehn quandle of a surface. Along the way, we prove that the involutory quotient of an Artin quandle is precisely the corresponding Coxeter quandle, and also determine the smallest nontrivial quotient of a braid quandle.

On insoluble transitive subgroups in the holomorph of a finite soluble group

A question of interest both in Hopf-Galois theory and in the theory of skew braces is whether the holomorph Hol(N) of a finite soluble group N can contain an insoluble regular subgroup. We investigate the more general problem of finding an insoluble transitive subgroup G in Hol(N) with soluble point stabilisers. We call such a pair (G,N) irreducible if we cannot pass to proper non-trivial quotients G‾, N‾ of G, N so that G‾ becomes a subgroup of Hol(N‾). We classify all irreducible solutions (G,N) of this problem, showing in particular that every non-abelian composition factor of G is isomorphic to the simple group of order 168. Moreover, every maximal normal subgroup of N has index 2.

On the structure of principal series representations of SL2(𝔽̄p)

Let [Formula: see text] be a prime number and [Formula: see text] be the algebraic closure of the finite field [Formula: see text] with [Formula: see text] elements. Let [Formula: see text] be an algebraically closed field of characteristic [Formula: see text]. Let [Formula: see text] and [Formula: see text] be the standard Borel subgroup of [Formula: see text]. For any character [Formula: see text] of [Formula: see text], define [Formula: see text], where [Formula: see text] is the group algebra of [Formula: see text]. In this paper, we prove two interesting properties of [Formula: see text]: (a) [Formula: see text] is either of finite length or residually finite; (b) Any nontrivial quotient of [Formula: see text] is finite dimensional.

Two-dimensional conformal field theory, full vertex algebra and current-current deformation

The main purpose of this paper is a mathematical construction of non-perturbative deformations of two-dimensional conformal field theories.We introduce the notion of a full vertex algebra which formulates a compact two-dimensional conformal field theory. Then, we construct a deformation family of full vertex algebras which serves as a current-current deformation of conformal field theory in physics. The parameter space of the deformations is expressed as a quotient of an orthogonal Grassmannian. This parameter space is a part of the CFT moduli space expected in physics. We demonstrate that nontrivial quotients of Grassmannians are obtained from holomorphic vertex operator algebras of central charge 24.

Iwahori–Hecke model for mod p representations of GL(2,F)

For a $p$-adic field $F$, the space of pro-$p$-Iwahori invariants of a universal supersingular mod $p$ representation $\tau$ of ${\rm GL}_2(F)$ is determined in the works of Breuil, Schein, and Hendel. The representation $\tau$ is introduced by Barthel and Livn\'e and this is defined in terms of the spherical Hecke operator. In earlier work of Anandavardhanan-Borisagar, an Iwahori-Hecke approach was introduced to study these universal supersingular representations in which they can be characterized via the Iwahori-Hecke operators. In this paper, we construct a certain quotient $\pi$ of $\tau$, making use of the Iwahori-Hecke operators. When $F$ is not totally ramified over $\mathbb Q_p$, the representation $\pi$ is a non-trivial quotient of $\tau$. We determine a basis for the space of invariants of $\pi$ under the pro-p Iwahori subgroup. A pleasant feature of this "new" representation $\pi$ is that its space of pro-$p$-Iwahori invariants admits a more uniform description vis-\`a-vis the description of the space of pro-$p$-Iwahori invariants of $\tau$.

A closer look at the non-Hopfianness of $BS(2,3)$

The Baumslag-Solitar group $BS(2,3)$, is a so-called non-Hopfian group, meaning that it has an epimorphism $\phi$ onto itself, that is not injective. In particular this is equivalent to saying that $BS(2,3)$ has a non-trivial quotient that is isomorphic to itself. As a consequence the Cayley graph of $BS(2,3)$ has a quotient that is isomorphic to itself up to change of generators. We describe this quotient on the graph-level and take a closer look at the most common epimorphism $\phi$. We show its kernel is a free group of infinite rank with an explicit set of generators. Finally we show how $\phi$ appears as a morphism on fundamental groups induced by some continuous map. This point of view was communicated to the author by Gilbert Levitt.

On the smallest non-trivial quotients of mapping class groups

We prove that the smallest non-trivial quotient of the mapping class group of a connected orientable surface of genus g \geq 3 without punctures is Sp _{2g}(2) , thus confirming a conjecture of Zimmermann. In the process, we generalise Korkmaz’s results on \mathbb C -linear representations of mapping class groups to projective representations over any field.

Restrictions on Homflypt and Kauffman polynomials arising from local moves

We study the effects of certain local moves on Homflyptand Kauffman polynomials. We show that all Homflypt(or Kauffman) polynomials are equal in a certain nontrivial quotient of the Laurent polynomial ring. As a consequence, we discover some new properties of these invariants.

Efficient Gröbner bases computation over principal ideal rings

In this paper we present new techniques for improving the computation of strong Gröbner bases over a principal ideal ring R. More precisely, we describe how to lift a strong Gröbner basis along a canonical projection R→R/n, n≠0, and along a ring isomorphism R→R1×R2. We then apply this to the computation of strong Gröbner bases over a non-trivial quotient of a principal ideal domain R/nR. The idea is to run a standard Gröbner basis algorithm pretending R/nR to be field. If we discover a non-invertible leading coefficient c, we use this information to try to split n=ab with coprime a,b. If this is possible, we recursively reduce the original computation to two strong Gröbner bases computations over R/aR and R/bR respectively. If no such c is discovered, the returned Gröbner basis is already a strong Gröbner basis for the input ideal over R/nR.

Orthogonality spaces of finite rank and the complex Hilbert spaces

An orthogonality space is a set endowed with a symmetric, irreflexive binary relation. By means of the usual orthogonality relation, each anisotropic quadratic space gives rise to such a structure. We investigate in this paper the question to which extent this strong abstraction suffices to characterize complex Hilbert spaces, which play a central role in quantum physics. To this end, we consider postulates concerning the nature and existence of symmetries. Together with a further postulate excluding the existence of nontrivial quotients, we establish a representation theorem for finite-dimensional orthomodular spaces over a dense subfield of [Formula: see text].

Every simple compact semiring is finite

A Hausdorff topological semiring is called simple if every non-zero continuous homomorphism into another Hausdorff topological semiring is injective. Classical work by Anzai and Kaplansky implies that any simple compact ring is finite. We generalize this result by proving that every simple compact semiring is finite, i.e., every infinite compact semiring admits a proper non-trivial quotient.

Infinite locally dihedral groups as automorphism groups

If \(A\) is a nontrivial torsion-free, locally cyclic group with no nontrivial divisible quotients, and \(G\) is the split extension of \(A\) by a group of order 2 acting on \(A\) by means of the inverting map, then \(G\simeq {{{\mathrm{Aut}}}G} \). We prove that in no other case the full automorphism group of a group is infinite and locally dihedral.

The L_2(11)-subalgebra of the Monster algebra

We study a subalgebra V of the Monster algebra, V M , generated by three Majorana axes a x , a y and a z indexed by the 2 A -involutions x , y and z of M , the Monster simple group. We use the notation V = << a x , a y , a z >> . We assume that x y is another 2 A -involution and that each of x z , y z and x y z has order 5 . Thus a subgroup G of M generated by { x , y , z } is a non-trivial quotient of the group G (5, 5, 5) = < x , y , z ∣ x 2 , y 2 , ( x y ) 2 , z 2 , ( x z ) 5 , ( y z ) 5 , ( x y z ) 5 > . It is known that G (5, 5, 5) is isomorphic to the projective special linear group L 2 (11) which is simple, so that G is isomorphic to L 2 (11) . It was proved by S. Norton that (up to conjugacy) G is the unique 2 A -generated L 2 (11) -subgroup of M and that K = C M ( G ) is isomorphic to the Mathieu group M 12 . For any pair { t , s } of 2 A -involutions, the pair of Majorana axes { a t , a s } generates the dihedral subalgebra << a t , a s >> of V M , whose structure has been described in . In particular, the subalgebra << a t , a s >> contains the Majorana axis a t s t by the conjugacy property of dihedral subalgebras. Hence from the structure of its dihedral subalgebras, V coincides with the subalgebra of V M generated by the set of Majorana axes { a t ∣ t ∈ T } , indexed by the 55 elements of the unique conjugacy class T of involutions of G ≅ L 2 (11) . We prove that V is 101 -dimensional, linearly spanned by the set { a t ⋅ a s ∣ s , t ∈ T } , and with C V M ( K ) = V ⊕ ι M , where ι M is the identity of V M . Lastly we present a recent result of Á. Seress proving that V is equal to the algebra of the unique Majorana representation of L 2 (11) .

Morphisms for resistive electrical networks

This paper presents a notion of morphism for resistive electrical networks. One can regard symmetries of a resistive network as morphisms, but they are trivial in the sense that they do not reduce the size of the problem. However, symmetries do induce a nontrivial quotient. We extend this result to arbitrary morphisms in a suitable categorical framework. Our principal application is to solve Kirchoff’s problem for certain products of paths.

Approximation theorems for group valued functions

Stone–Weierstrass-type theorems for groups of group-valued functions with a discrete range or a discrete domain are obtained. We study criteria for a subgroup of the group of continuous functions C ( X , G ) ( X compact, G a topological group) to be uniformly dense. These criteria are based on the existence of so-called condensing functions, where a continuous function ϕ : G → G is said to be condensing (respectively, finitely condensing) if it does not operate on any proper, point separating, closed subgroup of C ( K , G ) , with K compact, (respectively, with K finite) that contains the constant functions. The set D F ( G ) of finitely condensing functions in C ( G , G ) , is characterized, for any Abelian topological group G , as the set of those functions that are both non-affine and do not have nontrivial generalized periods (i.e. that do not factorize through nontrivial quotients of G ). This provides approximation theorems for functions with discrete domain and arbitrary (topological group) range. We also show that when G is discrete, every finitely condensing functions is condensing. The set of D ( G ) of condensing functions is thus characterized for discrete Abelian G . This provides approximation theorems for functions with an arbitrary (compact) domain and a discrete range. Answering an old question of Sternfeld, the description of D ( Z ) that follows is particularly simple: given ϕ : Z → Z , ϕ ∈ D ( Z ) if and only if for every k ∈ N with k ≥ 2 , there are n 1 , n 2 ∈ Z such that n 1 − n 2 is a multiple of k , while ϕ ( n 1 ) − ϕ ( n 2 ) is not.

Vanishing of trace forms in low characteristics

A finite-dimensional representation of an algebraic group G gives a trace symmetric bilinear form on the Lie algebra of G. We give a criterion in terms of the root system data for this form to vanish. As a corollary, we show that a Lie algebra of type E8 over a field of characteristic 5 does not have a so-called “quotient trace form”, answering a question posed in the 1960s. Let G be an algebraic group over a field F , acting on a finite-dimensional vector space V via a homomorphism ρ : G → GL(V ). The differential dρ of ρ maps the Lie algebra Lie(G) of G into gl(V ), and we put Trρ for the symmetric bilinear form Trρ(x, y) := trace(dρ(x) dρ(y)) for x, y ∈ Lie(G). We call Trρ a trace form of G. Such forms appear, for example, in the hypotheses for the Jacobson-Morozov Theorem [Ca, 5.3.1]. We prove: Theorem A. Assume G is simply connected, split, and almost simple. Then the following are equivalent: (a) The characteristic of F is a torsion prime for G. (b) Every trace form of G is zero. The set of torsion primes for G is given by the following table, cf. e.g. [St 75, 1.13]: type of G torsion primes An, Cn none Bn (n ≥ 3), Dn (n ≥ 4), G2 2 F4, E6, E7 2, 3 E8 2, 3, 5 A prime p is called a torsion prime for G if the corresponding group G(C) over C (or, equivalently, its compact form) is such that one of its homology groups, with coefficients in Z, contains an element of order p. We also prove a generalization of Theorem A that removes the hypotheses “simply connected” and “split”; it is somewhat more complicated, so we leave the statement until Th. D (and Remark 4.6). Replacing the simply connected group G with a nontrivial quotient G changes the situation in two ways: the group G has “fewer” representations and the Lie algebras of G and G may be different. These two changes are reflected in the integers N(G) and E(G) defined below. As a particular example of Th. A, for G of type E8 over a field of characteristic 2, 3, or 5, Trρ is zero for every representation ρ of G. One may ask whether the same is true for the representations of the Lie algebra Lie(G). That is, for a representation ψ of Lie(G), we write Trψ for the bilinear form (x, y) 7→ trace(ψ(x)ψ(y)), and ask 2000 Mathematics Subject Classification. 20G05 (17B50, 17B25).

INVOLUTIONS ON THE SECOND DUALS OF GROUP ALGEBRAS AND A MULTIPLIER PROBLEM

AbstractWe show that if a locally compact group $G$ is non-discrete or has an infinite amenable subgroup, then the second dual algebra $L^1(G)^{**}$ does not admit an involution extending the natural involution of $L^1(G)$. Thus, for the above classes of groups we answer in the negative a question raised by Duncan and Hosseiniun in 1979. We also find necessary and sufficient conditions for the dual of certain left-introverted subspaces of the space $C_b(G)$ (of bounded continuous functions on $G$) to admit involutions. We show that the involution problem is related to a multiplier problem. Finally, we show that certain non-trivial quotients of $L^1(G)^{**}$ admit involutions.

Detecting pro-p-groups that are not absolute Galois groups

Let p be a prime. It is a fundamental problem to classify the absolute Galois groups G F of fields F containing a primitive p th root of unity ξ p . In this paper we present several constraints on such G F , using restrictions on the cohomology of index p normal subgroups from N. Lemire, J. Mináč , and J. Swallow , Galois module structure of Galois cohomology and partial Euler-Poincaré characteristics, J. reine angew. Math. 613 (2007), 147–173. In section 1 we classify all maximal p -elementary abelian-by-order p quotients of these G F . In the case p > 2, each such quotient contains a unique closed index p elementary abelian subgroup. This seems to be the first case in which one can completely classify nontrivial quotients of absolute Galois groups by characteristic subgroups of normal subgroups. In section 2 we derive analogues of theorems of Artin-Schreier and Becker for order p elements of certain small quotients of G F . Finally, in section 3 we construct a new family of pro- p -groups which are not absolute Galois groups over any field F .

Satisfaction of existential theories in finitely presented groups and some embedding theorems

The main result is that for every recursively enumerable existential consistent theory Γ (in the usual language of group theory), there exists a finitely presented SQ-universal group H such that Γ is satisfied in every nontrivial quotient of H . Furthermore if Γ is satisfied in some group with a soluble word problem, then H can be taken with a soluble word problem. We characterize the finitely generated groups with soluble word problem as the finitely generated groups G for which there exists a finitely presented group H all of the nontrivial quotients of which embed G . We prove also that for every countable group G , there exists a 2-finitely generated SQ-universal group H such that every nontrivial quotient of H embeds G .