Semidirect Product Research Articles

Fixed point conditions for non-coprime actions

In the setting of finite groups, suppose $J$ acts on $N$ via automorphisms so that the induced semidirect product $N\rtimes J$ acts on some non-empty set $\Omega$ , with $N$ acting transitively. Glauberman proved that if the orders of $J$ and $N$ are coprime, then $J$ fixes a point in $\Omega$ . We consider the non-coprime case and show that if $N$ is abelian and a Sylow $p$ -subgroup of $J$ fixes a point in $\Omega$ for each prime $p$ , then $J$ fixes a point in $\Omega$ . We also show that if $N$ is nilpotent, $N\rtimes J$ is supersoluble, and a Sylow $p$ -subgroup of $J$ fixes a point in $\Omega$ for each prime $p$ , then $J$ fixes a point in $\Omega$ .

Muller–Speh–Vogan type irreducibility criterion for generalized principal series

We provide an elementary way to obtain an irreducibility criterion for generalized principal series of p-adic reductive groups and the finite central covering groups, extending known and frequently employed results for principal series. Basically our approach is a counting argument which rests on a semi-direct product decomposition of the relative Weyl group and its action on generalized principal series, in conjunction with the theory of Jacquet module. We thus circumvent the obstacle that the relative Weyl group of a general Levi subgroup is not a Coxeter group. Along the way, we observe a structural irreducibility criterion with respect to Levi subgroups. Indeed, following Speh–Vogan's argument for real groups which is essentially a corollary of intertwining operator theory, the former result can be generalized to parabolic inductions inducing from essentially tempered representations, but the latter one can't be obtained in this way. In view of the lack of intertwining operator theory for mod ℓ representations, our counting argument may play a role there, as opposed to the failure of Speh–Vogan's argument.

Profinite groups with abelian Sylow subgroups

We extend the definition of a finite -group to profinite groups and give a description of profinite -groups as a triple semidirect product of two prosoluble groups with a semisimple group, extending an old result of A. M. Broshi to the profinite case. We also prove that a profinite -group with finitely generated non-trivial Fitting subgroup is metabelian-by-(finite exponent). If, in addition, G is finitely generated then it is virtually metabelian polycyclic. Communicated by Mandi Schaeffer Fry

Quasi-Whittaker modules for the n-th Schrödinger algebra

The n-th Schrödinger algebra schn defined in [14] is the semi-direct product of the Lie algebra sl2 with the n-th Heisenberg Lie algebra hn, which generalizes the Schrödinger algebra sl2⋉h1. Let ϕ:hn→C be a nonzero Lie algebra homomorphism. A schn-module V is called quasi-Whittaker of type ϕ if V=U(schn)v, where U(schn) is the universal enveloping algebra of schn, v is a nonzero vector such that xv=ϕ(x)v for any x∈hn. In this paper, we prove that a simple schn-module V is a quasi-Whittaker module if and only if V is a locally finite hn-module. Then we classify the simple quasi-Whittaker modules of ϕ, according to the rank of ϕ. Furthermore, we characterize arbitrary quasi-Whittaker modules through the rank of ϕ.

The derivation algebra and automorphism group of the n-th Schrödinger algebra

The n-th Schrödinger algebra is the semidirect product of the special linear Lie algebra and the Heisenberg algebra . In this paper, we describe explicitly the structure of the derivation algebra and automorphism group of the n-th Schrödinger algebra. Especially, the automorphism group of the Schrödinger algebra is obtained.

Conjugacy languages in virtual graph products

In this paper we study the behaviour of conjugacy languages in virtual graph products, extending results by Ciobanu et al. [13]. We focus primarily on virtual graph products in the form of a semi-direct product. First, we study the behaviour of twisted conjugacy representatives in right-angled Artin and Coxeter groups. We prove regularity of the conjugacy geodesic language for virtual graph products in certain cases, and highlight properties of the spherical conjugacy language, depending on the automorphism and ordering on the generating set. Finally, we give a criterion for when the spherical conjugacy language is not unambiguous context-free for virtual graph products. We can extend this further in the case of virtual RAAGs, to show the spherical conjugacy language is not context-free.

Coding Matrices for Wreath Products of Groups

Wreath product, a powerful construction in group theory, has found extensive applications in various areas of mathematics and computer science. In this paper, we present a comprehensive analysis of coding matrices associated with wreath products. The coding matrices for the wreath product of two cyclic finite groups were given for the first time. It gives a generalization of the coding matrices for the semi-direct product. We found out that the coding matrix of wreath product really has the same shape as the one for semidirect product and gave the RW-matrix for the coding matrix. An example was showed to illustrate the assertions. Conditions were also given for different wreath products of cyclic groups and that gives different orders for the wreath products.

A study of the k-step generalized balancing sequences

In this paper, firstly, we define the k-step generalized Balancing sequences and study the Binet formula of these sequences. Also, we find families of super-diagonal matrices such that the permanents of these matrices are the elements of the k-step generalized Balancing sequences. Finally, we examine the periods of the k-step Balancing sequences in the semi-direct product presented by G = < x, y | x2m−1 = y2 = 1, yxy = x−1 > for the generating pair (x, y).

Clifford group is not a semidirect product in dimensions N divisible by four

The paper is devoted to projective Clifford groups of quantum N-dimensional systems (with configuration space ). Clearly, Clifford gates allow only the simplest quantum computations which can be simulated on a classical computer (Gottesmann–Knill theorem). However, it may serve as a cornerstone of full quantum computation. As to its group structure it is well-known that—in N-dimensional quantum mechanics—the Clifford group is a natural semidirect product provided the dimension N is an odd number. For even N special results on the Clifford groups are scattered in the mathematical literature, but they mostly do not concern the semidirect structure. Using appropriate group presentation of it is proved that for even N the projective Clifford groups are not natural semidirect products if and only if N is divisible by four.

Isomorphism classes of idempotent evolution algebras

We showed that isomorphism classes of idempotent evolution algebras are in bijection with the orbits of the semidirect product group of the symmetric group and the torus, considered the combinatoric problem of enumeration of isomorphism classes for these algebras over arbitrary finite fields, derived a general counting formula, and obtained explicit formulas for the numbers of isomorphism classes in dimensions 2, 3, and 4 over any finite field.

Substitution Principle and semidirect products

AbstractIn the classical theory of regular languages, the concept of recognition by profinite monoids is an important tool. Beyond regularity, Boolean spaces with internal monoids (BiMs) were recently proposed as a generalization. On the other hand, fragments of logic defining regular languages can be studied inductively via the so-called “Substitution Principle.” In this paper, we make the logical underpinnings of this principle explicit and extend it to arbitrary languages using Stone duality. Subsequently, we show how it can be used to obtain topo-algebraic recognizers for classes of languages defined by a wide class of first-order logic fragments. This naturally leads to a notion of semidirect product of BiMs extending the classical such construction for profinite monoids. Our main result is a generalization of Almeida and Weil’s Decomposition Theorem for semidirect products from the profinite setting to that of BiMs. This is a crucial step in a program to extend the profinite methods of regular language theory to the setting of complexity theory.

On character varieties with non-connected structure groups

For any connected complex reductive group G, any finitely generated discrete group Π and a normal subgroup Π˜ with quotient group Γ, we study the associated G⋊Γ-character variety, the space of admissible G⋊Γ-representations of Π. We study the relation between this variety and the Γ-fixed points in the usual G-character variety associated to Π˜. In the process, we give the classification of the isomorphism classes of the semi-direct products G⋊Γ, with fixed G and Γ. In the case where Π˜ and Π are fundamental groups of Riemann surfaces, a genericity condition on the conjugacy classes of monodromy is introduced as a sufficient condition for the irreducibility of G⋊Γ-representations. The example of GLn⋊<σ>-character varieties is discussed in detail.

Neutrino mass and mixing models with eclectic flavor symmetry ∆(27) ⋊ T′

The Kähler potentials of modular symmetry models receive unsuppressed contributions which may be controlled by a flavor symmetry, where the combination of the two symmetry types is referred to as eclectic flavor symmetry. After briefly reviewing the consistency conditions of eclectic flavor symmetry models, including those with generalised (g)CP, we perform a comprehensive bottom-up study of eclectic flavor symmetry models based on Ω(1) ≅ ∆(27) ⋊ T′, consisting of the flavor symmetry ∆(27) in a semi-direct product with the modular symmetry T′. The modular transformations of different ∆(27) multiplets are given by solving the consistency condition. The eight nontrivial singlets of ∆(27) are related by T′ modular symmetry, and they have to be present or absent simultaneously in any Ω(1) model. The most general forms of the superpotential and Kähler potential invariant under Ω(1) are discussed, and the corresponding fermion mass matrices are presented. Based on the eclectic flavor group Ω(1), two concrete lepton models which can successfully describe the experimental data of lepton masses and mixing parameters are constructed. For the two models without gCP, all six mixing parameters vary in small regions. A nearly maximal atmospheric mixing angle θ23 and Dirac CP phase δCP are obtained in the first model. After considering the compatible gCP symmetry and the assumption of mathfrak{R}tau = 0 in the first model, the μ − τ reflection symmetry is preserved in the charged lepton diagonal basis. As a consequence, the atmospheric mixing angle and Dirac CP phase are predicted to be maximal, and two Majorana CP phases are predicted to be π.

On split extensions of preordered groups

We investigate the behaviour of split extensions in the category OrdGrp of (pre)- ordered groups. Namely we show that the lexicographic order plays a key role on the existence of compatible orders for semidirect products, establishing necessary and sufficient conditions for such existence; we prove that the Split Short Five Lemma holds for (stably) strong split extensions, and identify classes of split extensions which admit a classifier.

The Fubini's Theorem for the Even Index Lie Subgroup

In this paper we present a new proof to generalize the Fubini's Theorem for the even index subgroup of a compact Lie group, using a recent result on the semi-direct product of groups. We also present some applications of this result for the invariant theory.

Sympathetic Lie algebras and adjoint cohomology for Lie algebras

We study sympathetic (i.e., perfect and complete) Lie algebras. Among other topics they arise in the study of adjoint Lie algebra cohomology. Here a motivation comes from a conjecture of Pirashvili, which says that a finite-dimensional complex perfect Lie algebra is semisimple if and only if its adjoint cohomology vanishes. We prove several general results for sympathetic Lie algebras and for the adjoint Lie algebra cohomology of arbitrary finite-dimensional Lie algebras in characteristic zero using a result of Hochschild and Serre. Moreover, for certain semidirect products we obtain explicit results for the adjoint cohomology.

Moore mixed graphs from Cayley graphs

A Moore ( r , z , k ) -mixed graph G has every vertex with undirected degree r , directed in- and out-degree z , diameter k , and number of vertices (or order) attaining the corresponding Moore bound M ( r , z , k ) for mixed graphs. When the order of G is close to M ( r , z , k ) vertices, we refer to it as an almost Moore graph. The first part of this paper is a survey about known Moore (and almost Moore) general mixed graphs that turn out to be Cayley graphs. Then, in the second part of the paper, we give new results on the bipartite case. First, we show that Moore bipartite mixed graphs with diameter three are distance-regular, and their spectra are fully characterized. In particular, an infinity family of Moore bipartite (1, z , 3) -mixed graphs is presented, which are Cayley graphs of semidirect products of groups. Our study is based on the line digraph technique, and on some results about when the line digraph of a Cayley digraph is again a Cayley digraph.

Equivariant Observers for Second-Order Systems on Matrix Lie Groups

This paper develops an equivariant symmetry for second order kinematic systems on matrix Lie groups and uses this symmetry for observer design. The state of a second order kinematic system on a matrix Lie group is naturally posed on the tangent bundle of the group with the inputs lying in the tangent of the tangent bundle known as the double tangent bundle. We provide a simple parameterization of both the tangent bundle state space and the input space (the fiber space of the double tangent bundle) and then introduce a semi-direct product group and group actions onto both the state and input spaces. We show that with the proposed group actions the second order kinematics are equivariant. An equivariant lift of the kinematics onto the symmetry group is derived and used to design nonlinear observers on the lifted state space using nonlinear constructive design techniques. The observer design is specialised to kinematics on groups that themselves admit a semi-direct product structure and include applications in rigid-body motion amongst others. A simulation based on an ideal hovercraft model verifies the performance of the proposed observer architecture.

On amenable Hilbert-Schmidt stable groups

We examine Hilbert-Schmidt stability (HS-stability) of discrete amenable groups from several angles. We give a short, elementary proof that finitely generated nilpotent groups are HS-stable. We investigate the permanence of HS-stability under central quotients by showing HS-stability is preserved by finite central quotients, but is not preserved in general. We give a characterization of HS-stability for semidirect products G⋊γZ with G abelian. We use it to construct the first example of a finitely generated amenable HS-stable group which is not permutation stable. Finally, it is proved that for amenable groups flexible HS-stability is equivalent to HS-stability, and very flexible HS stability is equivalent to maximal almost periodicity. There is some overlap of our work with the very recent and very nice preprint [27] of Levit and Vigdorovich. We detail this overlap in the introduction. Where our work overlaps it appears that we take different approaches to the proofs and we feel the two works complement each other.

Constructing directed strongly regular graphs by using semidirect products and semidihedral groups

In this paper, directed strongly regular graphs (DSRGs) are constructed by using semidirect products. The orbit condition in [3] has been weakened and this gives rise to the construction of DSRGs. Moreover, a different construction is given for DSRG by using semidihedral groups.