Group Aut Research Articles

ISOMETRY GROUPS OF GENERALIZED STIEFEL MANIFOLDS

A generalized Stiefel manifold is the manifold of orthonormal frames in a vector space with a nondegenerated bilinear or hermitian form. In this article, the Isometry group of the generalized Stiefel manifolds are computed at least up to connected components in an explicit form. This is done by considering a natural nonassociative algebra m associated to the affine structure of the Stiefel manifold, computing explicitly the automorphism group Aut(𝔪) and inducing this computation to the Isometry group.

On the classification of binary completely transitive codes with almost-simple top-group

A code C in the Hamming metric, that is, is a subset of the vertex set VΓ of the Hamming graph Γ=H(m,q), gives rise to a natural distance partition{C,C1,…,Cρ}, where ρ is the covering radius of C. Such a code C is called completely transitive if the automorphism group Aut(C) acts transitively on each of the sets C, C1, …, Cρ. A code C is called 2-neighbour-transitive if ρ⩾2 and Aut(C) acts transitively on each of C, C1 and C2.Let C be a completely transitive code in a binary (q=2) Hamming graph having full automorphism group Aut(C) and minimum distance δ⩾5. Then it is known that Aut(C) induces a 2-homogeneous action on the coordinates of the vertices of the Hamming graph. The main result of this paper classifies those C for which this induced 2-homogeneous action is not an affine, linear or symplectic group. We find that there are 13 such codes, 4 of which are non-linear codes. Though most of the codes are well-known, we obtain several new results. First, a non-linear completely transitive code that does not explicitly appear in the existing literature is constructed, as well as a related non-linear code that is 2-neighbour-transitive but not completely transitive. Moreover, new proofs of the complete transitivity of several codes are given. Additionally, we consider the question of the existence of distance-regular graphs related to the completely transitive codes appearing in our main result.

The groups Aut and Out of the fundamental group of a closed Sol 3-manifold

Let [Formula: see text] be the fundamental group of a closed [Formula: see text] [Formula: see text]-manifold. We describe the groups [Formula: see text] and [Formula: see text]. We first consider the case where [Formula: see text] is the fundamental group of a torus bundle, and then the case where [Formula: see text] is the fundamental group of a closed [Formula: see text] [Formula: see text]-manifold which is not a torus bundle. The groups are described in terms of some iterated semidirect products of well-known groups, where one exception is for [Formula: see text] for some [Formula: see text], in which case [Formula: see text] is described as an extension and a presentation is given.

Geometric Structures Generated by the Same Dynamics. Recent Results and Challenges

The main contribution of this review is to show some relevant relationships between three geometric structures on a connected Lie group G, generated by the same dynamics. Namely, Linear Control Systems, Almost Riemannian Structures, and Degenerate Dynamical Systems. These notions are generated by two ordinary differential equations on G: linear and invariant vector fields. A linear vector field on G is determined by its flow, a 1-parameter group of Aut(G), the Lie group of G-automorphisms. An invariant vector field is just an element of the Lie algebra g of G. The Jouan Equivalence Theorem and the Pontryagin Maximum Principal are instrumental in this setup, allowing the extension of results from Lie groups to arbitrary manifolds for the same kind of structures which satisfy the Lie algebra finitude condition. For each structure, we present the first given examples; these examples generate the systems in the plane. Next, we introduce a general definition for these geometric structures on Euclidean spaces and G. We describe recent results of the theory. As an additional contribution, we conclude by formulating a list of open problems and challenges on these geometric structures. Since the involved dynamic comes from algebraic structures on Lie groups, symmetries are present throughout the paper.

On the automorphism group of a binary form associated with algebraic trigonometric quantities

TextLet F(x,y) be a binary form of degree at least three and non-zero discriminant. In this article we compute the automorphism group Aut F for four families of binary forms. The first two families that we are interested in are homogenizations of minimal polynomials of 2cos⁡(2πn) and 2sin⁡(2πn), which we denote by Ψn(x,y) and Πn(x,y), respectively. The remaining two forms that we consider are homogenizations of Chebyshev polynomials of first and second kinds, denoted Tn(x,y) and Un(x,y), respectively. VideoFor a video summary of this paper, please visit https://youtu.be/1Ge-cm4ilek.

On n-partite digraphical representations of finite groups

A group G admits an n-partite digraphical representation if there exists a regular n-partite digraph Γ such that the automorphism group Aut(Γ) of Γ satisfies the following properties:(1)Aut(Γ) is isomorphic to G,(2)Aut(Γ) acts semiregularly on the vertices of Γ and(3)the orbits of Aut(Γ) on the vertex set of Γ form a partition into n parts giving a structure of n-partite digraph to Γ.In this paper, for every positive integer n, we classify the finite groups admitting an n-partite digraphical representation.

Out-of-season breeding and ewe-lamb bond from birth to weaning in Corriedale sheep

Out-of-season lambing is based in productive aims; however also modifies the basic physiology of the ewe. It might affect ewes and lambs’ behaviours at birth, their bond, the process of lambs’ independence from their mothers and their response to weaning. This study aimed to compare the ewe-lamb bond from birth to artificial weaning in out-of-season lambing (autumn) and spring lambing. A complementary aim was to determine if out-of-season breeding affects ewes' udder size, milk yield and composition, body condition score (BCS) and body weight of the ewes and lambs. The study was performed with 26 multiparous single-lambing Corriedale ewes that lambed in spring (natural season: group SPR), and 26 ewes that lambed in autumn (out-of-season: group AUT) and their lambs. The ewe-lamb behaviours were recorded during a separation-reunion test performed 24–36 h after birth. Lambs’ behaviours were recorded before and after weaning (at 80 days of age). Ewes’ BCS and body weight were recorded at mating, lambing and weaning. Lambs’ body weight was recorded at lambing, before and after weaning. Ewes’ milk yield and composition were determined before weaning. During the separation-reunion test, although AUT lambs vocalised more often than SPR lambs their mothers remained further away (P = 0.01). Before weaning, AUT ewe-lamb dyads were observed more often far from each other (P < 0.001); AUT lambs attempted to suckle more times (P < 0.001) and grazed less frequently than SPR lambs (P = 0.004). At weaning, AUT lambs walked more frequently, but were less often standing up than SPR lambs (P < 0.001 for both behaviours). While SPR lambs decreased their grazing frequency on the day of weaning, AUT lambs increased it, grazing more frequently than SPR lambs (P < 0.001). AUT ewes lambed with a greater BCS and body weight than SPR ewes (P = 0.01 and P = 0.031, respectively), but this was reverted at weaning (BCS: P < 0.001; body weight: P = 0.002). SPR ewes had a greater milk yield, with more fat, lactose and protein amount than AUT ewes (P < 0.001 for all variables). At weaning, SPR lambs were heavier than AUT lambs (P < 0.001). In conclusion, although AUT ewes were in better body condition after birth, the ewe-lamb bond of the AUT group was weaker. This was associated with a lower expression of maternal attention when AUT lambs requested maternal care. The better environmental conditions of SPR dyads during lactation might positively impact their social and nutritional behaviours. The difference between groups remained until weaning, so AUT and SPR lambs displayed different behavioural strategies to cope with artificial weaning.

Isolated Joint Block Progression Training Improves Leaping Performance in Dancers.

The purpose of this study was to investigate the effect of a 12-week ankle-specific block progression training program on saut de chat leaping performance [leap height, peak power (PP), joint kinetics and kinematics], maximal voluntary isometric plantar flexion (MVIP) strength, and Achilles tendon (AT) stiffness. Dancers (training group n = 7, control group n = 7) performed MVIP at plantarflexed (10◦) and neutral ankle positions (0◦) followed by ramping isometric contractions equipped with ultrasound to assess strength and AT stiffness, respectively. Dancers also performed saut de chat leaps surrounded by 3-D motion capture atop force platforms to determine center of mass and joint kinematics and kinetics. The training group then followed a 12-week ankle-focused program including isometric, dynamic constant external resistance, accentuated eccentric loading, and plyometric training modalities, while the control group continued dancing normally. We found that the training group's saut de chat ankle PP (59.8%), braking ankle stiffness (69.6%), center of mass PP (11.4%), and leap height (12.1%) significantly increased following training. We further found that the training group's MVIP significantly increased at 10◦ (17.0%) and 0◦ (12.2%) along with AT stiffness (29.6%), while aesthetic leaping measures were unchanged (peak split angle, mean trunk angle, trunk angle range). Ankle-specific block progression training appears to benefit saut de chat leaping performance, PP output, ankle-joint kinetics, maximal strength, and AT stiffness, while not affecting kinematic aesthetic measures. We speculate that the combined training blocks elicited physiological changes and enhanced neuromuscular synchronization for increased saut de chat leaping performance in this cohort of dancers.

On the Monoid of Unital Endomorphisms of a Boolean Ring

Let X be a nonempty set and P(X) the power set of X. The aim of this paper is to provide an explicit description of the monoid End1P(X)(P(X)) of unital ring endomorphisms of the Boolean ring P(X) and the automorphism group Aut(P(X)) when X is finite. Among other facts, it is shown that if X has cardinality n≥1, then End1P(X)(P(X))≅Tnop, where Tn is the full transformation monoid on the set X and Aut(P(X))≅Sn.

Symmetries of complex analytic vector fields with an essential singularity on the Riemann sphere

Abstract We consider the family ℰ (s, r, d) of all singular complex analytic vector fields X ( z ) = Q ( z ) P ( z ) e E ( z ) ∂ ∂ z $X(z)=\frac{Q(z)}{P(z)}{{e}^{E(z)}}\frac{\partial }{\partial z}$ on the Riemann sphere where Q, P, ℰ are polynomials with deg Q = s, deg P = r and deg ℰ = d ≥ 1. Using the pullback action of the affine group Aut(ℂ) and the divisors for X, we calculate the isotropy groups Aut(ℂ) X of discrete symmetries for X ∈ ℰ (s, r, d). The subfamily ℰ (s, r, d)id of those X with trivial isotropy group in Aut(ℂ) is endowed with a holomorphic trivial principal Aut(ℂ)-bundle structure. A necessary and sufficient arithmetic condition on s, r, d ensuring the equality ℰ (s, r, d) = ℰ (s, r, d)id is presented. Moreover, those X ∈ ℰ (s, r, d) \ ℰ (s, r, d)id with non-trivial isotropy are realized. This yields explicit global normal forms for all X ∈ ℰ (s, r, d). A natural dictionary between analytic tensors, vector fields, 1-forms, orientable quadratic differentials and functions on Riemann surfaces M is extended as follows. In the presence of nontrivial discrete symmetries Γ &lt; Aut(M), the dictionary describes the correspondence between Γ-invariant tensors on M and tensors on M /Γ.

"An algorithm for automorphisms of infinite dimensional Grassmann algebras"

"Let G be the infinite dimensional Grassmann algebra. In this study, we determine a subgroup of the automorphism group Aut(G) of the algebra G which is of an importance in the description of the group Aut(G). We give an infinite generating set for this subgroup and suggest an algorithm which shows how to express each automorphism as compositions of generating elements."

A characterization of Johnson and Hamming graphs and proof of Babai's conjecture

One of the central results in the representation theory of distance-regular graphs classifies distance-regular graphs with μ≥2 and second largest eigenvalue θ1=b1−1. In this paper we give a classification under the (weaker) approximate eigenvalue constraint θ1≥(1−ε)b1 for the class of geometric distance-regular graphs. As an application, we confirm Babai's conjecture on the minimal degree of the automorphism group of distance-regular graphs of bounded diameter. This conjecture asserts that if X is a primitive distance-regular graph with n vertices, and X is not a Hamming graph or a Johnson graph, then the automorphism group Aut(X) has minimal degree ≥cn for some constant c>0. It follows that Aut(X) satisfies strong structural constraints.

On spherical unitary representations of groups of spheromorphisms of Bruhat–Tits trees

Consider an infinite homogeneous tree \mathcal{T}_n of valence n+1 , its group Aut (\mathcal{T}_n) of automorphisms, and the group Hier (\mathcal{T}_n) of its spheromorphisms (hierarchomorphisms), i.e., the group of homeomorphisms of the boundary of \mathcal{T}_n that locally coincide with transformations defined by automorphisms. We show that the subgroup Aut (\mathcal{T}_n) is spherical in Hier (\mathcal{T}_n) , i.e., any irreducible unitary representation of Hier (\mathcal{T}_n) contains at most one Aut (\mathcal{T}_n) -fixed vector. We present a combinatorial description of the space of double cosets of Hier (\mathcal{T}_n) with respect to Aut (\mathcal{T}_n) and construct a “new” family of spherical representations of Hier (\mathcal{T}_n) . We also show that the Thompson group Th has PSL (2,\mathbb{Z}) -spherical unitary representations.

Covering groups of M22 as regular Galois groups over [formula omitted

We close a gap in the literature by showing that the full covering groups of the Mathieu group M22 and of its automorphism group Aut(M22) occur as the Galois groups of Q-regular Galois extensions of Q(t).

On representations of [formula omitted], [formula omitted] and [formula omitted

By work of Belyĭ [2], the absolute Galois group GQ=Gal(Q‾/Q) of the field Q of rational numbers can be embedded into A=Aut(Fˆ2), the automorphism group of the free profinite group Fˆ2 on two generators. The image of GQ lies inside GTˆ, the Grothendieck-Teichmüller group. While it is known that every abelian representation of GQ can be extended to GTˆ, Lochak and Schneps [13] put forward the challenge of constructing irreducible non-abelian representations of GTˆ. We do this virtually, namely by showing that a rich class of arithmetically defined representations of GQ can be extended to finite index subgroups of GTˆ. This is achieved, in fact, by extending these representations all the way to finite index subgroups of A=Aut(Fˆ2). We do this by developing a profinite version of the work of Grunewald and Lubotzky [7], which provided a rich collection of representations for the discrete group Aut(Fd).

On Automorphism Groups of Finite Chain Rings

A finite ring with an identity is a chain ring if its lattice of left ideals forms a unique chain. Let R be a finite chain ring with invaraints p,n,r,k,k′,m. If n=1, the automorphism group Aut(R) of R is known. The main purpose of this article is to study the structure of Aut(R) when n&gt;1. First, we prove that Aut(R) is determined by the automorphism group of a certain commutative chain subring. Then we use this fact to find the automorphism group of R when p∤k. In addition, Aut(R) is investigated under a more general condition; that is, R is very pure and p need not divide k. Based on the j-diagram introduced by Ayoub, we were able to give the automorphism group in terms of a particular group of matrices. The structure of the automorphism group of a finite chain ring depends essentially on its invaraints and the associated j-diagram.

The Hurwitz curve over a finite field and its Weierstrass points for the morphism of lines

For any smooth Hurwitz curve Hn:XYn+YZn+XnZ=0 over the finite field Fp, an explicit description of its Weierstrass points for the morphism of lines is presented. As a consequence, bounds on the number of rational points of Hn are obtained via Stöhr-Voloch Theory. Further, the full automorphism group Aut(Hn), as well as the genera of all Galois subcovers of Hn, with n≠3,pr, are computed. Finally, a question by F. Torres on plane nonsingular maximal curves is answered.

Regular Cayley maps for dihedral groups

An orientably-regular map M is a 2-cell embedding of a finite connected graph in a closed orientable surface such that the group Aut∘M of orientation-preserving automorphisms of M acts transitively on the set of arcs. Such a map M is called a Cayley map for the finite group G if Aut∘M contains a subgroup, which is isomorphic to G and acts regularly on the set of vertices. Conder and Tucker (2014) classified the regular Cayley maps for finite cyclic groups, and obtain two two-parameter families M(n,r), one for odd n and one for even n, where n is the order of the regular cyclic group and r is a positive integer satisfying certain arithmetical conditions. In this paper, we classify the regular Cayley maps for dihedral groups in the same fashion. Five two-parameter families Mi(n,r), 1≤i≤5, are derived, where 2n is the order of the regular dihedral group and r is an integer satisfying certain arithmetical conditions. For each map Mi(n,r), we determine its valence and covalence, and also describe the structure of the group Aut∘Mi(n,r). Unlike the approach of Conder and Tucker, which is entirely algebraic, we follow the traditional combinatorial representation of Cayley maps, and use a combination of permutation group theoretical techniques, the method of quotient Cayley maps, and computations with skew morphisms.

Spectral invariants for finite dimensional Lie algebras

For a Lie algebra L with basis {x1,x2,…,xn}, its associated characteristic polynomial QL(z) is the determinant of the linear pencil z0I+z1adx1+⋯+znadxn. This paper shows that QL is invariant under the automorphism group Aut(L). The zero variety and factorization of QL reflect the structure of L. In the case L is solvable QL is known to be a product of linear factors. This fact gives rise to the definition of spectral matrix and the Poincaré polynomial for solvable Lie algebras. Application is given to 1-dimensional extensions of nilpotent Lie algebras.

Del Pezzo surfaces over finite fields

Let X be a del Pezzo surface of degree 2 or greater over a finite field Fq. The image Γ of the Galois group Gal(F‾q/Fq) in the group Aut(Pic(X‾)) is a cyclic subgroup preserving the anticanonical class and the intersection form. The conjugacy class of Γ in the subgroup of Aut(Pic(X‾)) preserving the anticanonical class and the intersection form is a natural invariant of X. We say that the conjugacy class of Γ in Aut(Pic(X‾)) is the type of a del Pezzo surface. In this paper we study which types of del Pezzo surfaces of degree 2 or greater can be realized for given q. We collect known results about this problem and fill the gaps.