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Let $\mathrm{SL}_{n}(\mathbb{Z})$ $(n\geq 3)$ be the special linear group and $M^{r}$ be a closed aspherical manifold. It is proved that when $r, Comment: The article arXiv:1609.07699 is split into two parts. This is the second part. The title is also changed. To appear in Algebraic and Geometric Topology

We present a practical cryptanalysis of WalnutDSA, a digital signature algorithm trademarked by SecureRF. WalnutDSA uses techniques from permutation groups, matrix groups and braid groups, and is designed to provide post-quantum security in lightweight IoT device contexts. The attack given in this paper bypasses the E-Multiplication\(^{\text {TM}}\) and cloaked conjugacy search problems at the heart of the algorithm and forges signatures for arbitrary messages in approximately two minutes. We also discuss potential countermeasures to the attack.

Let $p_n$ be the characteristic polynomial of an $n \times n$ random matrix drawn from one of the compact classical matrix groups. We show that the critical points of $p_n$ converge to the uniform distribution on the unit circle as $n$ tends to infinity. More generally, we show the same limit for a class of random polynomials whose roots lie on the unit circle. Our results extend the work of Pemantle-Rivin and Kabluchko to the setting where the roots are neither independent nor identically distributed.

An analog of the Freudenthal–Weil theorem holds for the discontinuous homomorphisms of a connected pro-Lie group into a compact group if and only if the radical of the pro-Lie group is amenable.

Let be the ring of all matrices over a field . It is proved that a map is additive if and only if for all , the set of all invertible matrices over .

If every element of a matrix group is similar to a permutation matrix, then it is called a permutation-like matrix group. Refs. [3] and [4] showed that, if a permutation-like matrix group contains a maximal cycle of length equal to a prime or a square of a prime and the maximal cycle generates a normal subgroup, then it is similar to a permutation matrix group. In this paper, we prove that if a permutation-like matrix group contains a maximal cycle of length equal to any power of any odd prime and the maximal cycle generates a normal subgroup, then it is similar to a permutation matrix group.

We describe an algorithm to compute a composition tree for a matrix group defined over a finite field, and show how to use the associated structure to carry out computations with such groups; these include finding composition and chief series, the soluble radical, and Sylow subgroups.

It is shown that there exist d, ℓ ≥ 3 and a sequence C1, …, Cℓ of cyclic subgroups of the d-dimensional unitriangluar matrix group over ℤ (which is finitely generated nilpotent) such that membership in the product C1 C2 ⋯ Cℓ is undecidable.

We investigate the commutators of elements of the group of infinite unitriangular matrices over an associative ring with and a commutative group of invertible elements. We prove that every unitriangular matrix of a specified form is a commutator of two other unitriangular matrices. As a direct consequence, we give a complete characterization of the lower central series of the group including the width of its terms with respect to basic commutators and Engel words. With an additional restriction on the ring , we show that the derived subgroup of coincides with the group . These results generalize the results obtained for triangular groups over a field.

Let R denote the Galois ring of characteristic p s and cardinality p s h . In this paper, we determine the Smith normal forms of matrices over R , compute the number of the orbits of m × n matrices under the group GL m ( R ) × GL n ( R ) and the length of each orbit. Moreover, we discuss their applications to association schemes, systems of linear equations, generalized invertible matrices, idempotent matrices and involutory matrices.

A dominance rule for group invertible matrices using proper splitting is proposed, and used this notion to show that a matrix is group monotone. Then some possible applications are discussed.

We define pseudomatrix multiplication on the set of rectangular matrices which is different from the standard matrix multiplication. We give examples of pseudomatrix groups. The usual matrix multiplication is a particular case of pseudomatrix multiplication.

The classification of a block-transitive designs is an important subject on algebraic combinatorics. With the aid of MATLAB software, using the classification theorem of 3-homogeneous permutation groups, we look at the classification problem of block-transitive 7–(v, k, 3) design and prove our main theorem: If the automorphism group of a 7–(v, k, 3) design is block-transitive, then it is neither isomorphic to Affine Type Groups nor Almost Simple Type Groups.

This article considers the work done to find a suitable group implementation for the creation of new cryptosystems which work is based on the use of finite non-commutative algebraic groups. In particular, some subgroup structures of matrix groups over finite fields are described, and one of the possible methods for representing of open texts as elements of the group is used. At the preliminary stage of the cryptosystem, the developed concept of generating keys is described, and recommendations are given for selecting the elements of the secret key. The structure and features of the cryptosystem under investigation are given, earlier studies and experiments that were carried out in this field are considered. To generate subgroups of the groups under study, it is proposed to use elementary transvections, in particular, long and short root unipotent elements. A classification of certain groups of the Chevalley type by types A l -D l is provided. Some assumptions are made concerning the possible improvement of reliability and speed of such cryptosystems based upon the finite non-commutative algebraic groups, as well as relating to the prospects of further research in this direction and minimizing or eliminating the shortcomings of this data encryption model.

Let $R$ be an infinite commutative ring with identity and $n\geq 2$ be an integer. We prove that for each integer $i=0,1,\cdots ,n-2,$ the $L^{2}$-Betti number $b_{i}^{(2)}(G)=0,$ $\ $when $G=\mathrm{GL}_{n}(R)$ the general linear group, $\mathrm{SL}_{n}(R)$ the special linear group, $% E_{n}(R)$ the group generated by elementary matrices. When $R$ is an infinite principal ideal domain, similar results are obtained for $\mathrm{Sp}_{2n}(R)$ the symplectic group, $\mathrm{ESp}_{2n}(R)$ the elementary symplectic group, $\mathrm{O}(n,n)(R)$ the split orthogonal group or $\mathrm{EO}(n,n)(R)$ the elementary orthogonal group. Furthermore, we prove that $G$ is not acylindrically hyperbolic if $n\geq 4$. We also prove similar results for a class of noncommutative rings. The proofs are based on a notion of $n$-rigid rings., 17 pages, to appear in Algebraic & Geometric Topology

In this chapter, we give two applications of Riemannian submersions and it consists of two sections. In the first section, we give an application of Riemannian submersions in robotic theory [12]. To this aim, we first define Lie groups and give basic properties of such manifolds, then we give applications of Riemannian submersions in redundant robotic theory. Applications of Riemannian submersions in Kaluza-Klein theory have been given in detail in the book [111] and papers [112] and [40]; therefore, in the second section of this chapter, we give brief information for the Kaluza-Klein theory in terms of principal fiber bundles and Riemannian submersions.

In this chapter we continue our introduction to matrix theory, beginning with the notion of a matrix inverse and the definition of a matrix group. For now, the main example of a matrix group is the group \(GL(n,\mathbb {F})\) of invertible \(n\times n\) matrices over a field \(\mathbb {F}\) and its subgroups. We will also show that every matrix \(A\in \mathbb {F}^{n\times n}\) can be factored as a product \({ LPDU}\), where each of L, P, D, and U is a matrix in an explicit subset of \(\mathbb {F}^{n\times n}\). For example, P is a partial permutation matrix, D is diagonal, and L and U are lower and upper triangular respectively.

Analytic properties of matrix groups and their Lie algebras are studied via the exponential and logarithm maps, following von Neumann.

Using an algebraic approach based on the theory of Coxeter groups, we design, and describe the performance of, a class of line codes for parallel transmission of $b$ bits over $b+1$ wires that admit especially simple encoding and decoding algorithms. A number of designs are exhibited, some of them being novel or improving on previously obtained codes., Comment: 19 pages, 10 figures

The two main sources of groups are the permutation groups and the matrix groups. This chapter is devoted to introduce these groups, and to study some of their fundamental and elementary properties.

Locally compact groups are studied more systematically, Haar measure is constructed, and the equivalence of various notions of amenability is shown. Property (T) is introduced, and proved for some important examples. Property (T) for a specific matrix group is used to exhibit an explicit family of expander graphs.

A unified treatment of the $2 \times 2$ analog of the Freudenthal-Tits magic square of Lie groups is given, providing an explicit representation in terms of matrix groups over composition algebras.

Extended regions located at an angle of 20° to the rolling plane are observed inside deformation bands in a (110)[001] Fe-3%Si alloy single crystal at a high strain (∼60%). These regions were interpreted earlier as shear bands. The lattice orientation in these bands is close to (110)[001], and their habit plane is parallel to the {112} planes of the deformed {111}〈112〉 matrix. The misorientations between the bands and the matrix group around special misorientations Σ9, Σ19a, Σ27a, and Σ33a, which are characterized by close angles of rotation about axis 〈110〉. During primary recrystallization, the (110)[001] grains growing from the bands retain segments of the corresponding special boundaries with the deformed matrix.

We express the Whitehead group K 1 of an arbitrary generalized matrix ring in terms of the K 1 group of the original rings.

In this paper, we focus on the following coupled linear matrix equations $$\begin{aligned} \mathcal {M}_i(X,Y)={\mathcal {M}_{i1}(X)+\mathcal {M}_{i2}(Y)}=L_i, \end{aligned}$$ with $$\begin{aligned} {\mathcal {M}_{i \ell }(W)}&= \sum \limits _{j = 1}^{q } \left( {{\sum \limits _{\lambda = 1}^{t_1^{(\ell )} } {A_{ij\lambda }^{(\ell )} } } W_j B^{(\ell )}_{ij\lambda } + {\sum \limits _{\mu = 1}^{t_2^{(\ell )} } {C_{ij\mu }^{(\ell )} \overline{W} _j D^{(\ell )}_{ij\mu } } } + {\sum \limits _{\nu = 1}^{t_3^{(\ell )} } {E^{(\ell )}_{ij\nu } W_{j}^T F^{(\ell )}_{ij\nu } } }}\right) , \\&\ell =1,2. \end{aligned}$$ where $$A^{(\ell )}_{ij\lambda },B^{(\ell )}_{ij\lambda }$$ , $$C^{(\ell )}_{ij\mu }, D^{(\ell )}_{ij\mu }$$ , $$E^{(\ell )}_{ij\nu },F^{(\ell )}_{ij\nu }$$ and $$L_i$$ (for $$i \in I[1,p]$$ ) are given matrices with appropriate dimensions defined over complex number field. Our object is to obtain the solution groups $$X=(X_1,X_2,\ldots ,X_q)$$ and $$Y=(Y_1,Y_2,\ldots ,Y_q)$$ of the considered coupled linear matrix equations such that $$X$$ and $$Y$$ are the groups of the Hermitian reflexive and skew-Hermitian matrices, respectively. To do so, an iterative algorithm is proposed which stops within finite number of steps in the exact arithmetic. Moreover, the algorithm determines the solvability of the mentioned coupled linear matrix equations over the Hermitian reflexive and skew-Hermitian matrices, automatically. In the case that the coupled linear matrix equations are consistent, the least-norm Hermitian reflexive and skew-Hermitian solution groups can be computed by choosing suitable initial iterative matrix groups. In addition, the unique optimal approximate Hermitian reflexive and skew-Hermitian solution groups to given arbitrary matrix groups are derived. Finally, some numerical experiments are reported to illustrate the validity of our established theoretical results and feasibly of the presented algorithm.

Let R be a commutative ring of characteristic zero and G be an arbitrary group. In the present paper we classify the groups G for which the set of symmetric elements with respect to the classical involution of the group ring RG is Lie metabelian.

A matrix group is said to be permutation-like if any matrix of the group is similar to a permutation matrix. G. Cigler proved that, if a permutation-like matrix group contains a normal cyclic subgroup which is generated by a maximal cycle and the matrix dimension is a prime, then the group is similar to a permutation matrix group. This paper extends the result to the case where the matrix dimension is a square of a prime.

We develop a new method for analyzing the large-N limit of matrix models. We assign the concept of energy to each matrix element and integrate the most highest energy to get a new matrix model which has reduced rank. By regarding this procedure as a renormalization group, we deduce the critical exponents in the large-N limit by elaborating fixed points of renormalization group transformation. For consistency of our method, we compare our result to that obtained by another method and find nice agreement.

We study products of matrices of fixed orders. We show that if g is an upper triangular matrix, finite or infinite, over a field of q elements, then g can be expressed as a product of at most four triangular matrices whose orders are divisors of q − 1 . This result can be applied to the general linear and to the Vershik–Kerov group. We also present some facts about conjugacy of elements of orders dividing q − 1 .

The purpose in this paper is to complete the classification of primitive sharp permutation groups of type ({0, k}, n) by proving that no such group can be almost simple.

Memoryless computation is a novel means of computing any function of a set of registers by updating one register at a time while using no memory. We aim to emulate how computations are performed on modern cores, since they typically involve updates of single registers. The computation model of memoryless computation can be fully expressed in terms of transformation semigroups, or in the case of bijective functions, permutation groups. In this paper, we view registers as elements of a finite field and we compute linear permutations without memory. We first determine the maximum complexity of a linear function when only linear instructions are allowed. We also determine which linear functions are hardest to compute when the field in question is the binary field and the number of registers is even. Secondly, we investigate some matrix groups, thus showing that the special linear group is internally computable but not fast. Thirdly, we determine the smallest set of instructions required to generate the special and general linear groups. These results are important for memoryless computation, for they show that linear functions can be computed very fast or that very few instructions are needed to compute any linear function. They thus indicate new advantages of using memoryless computation.

The automorphism group of a flag-transitive 6–(v, k, 2) design is a 3-homogeneous permutation group. Therefore, using the classification theorem of 3–homogeneous permutation groups, the classification of flag-transitive 6-(v, k,2) designs can be discussed. In this paper, by analyzing the combination quantity relation of 6–(v, k, 2) design and the characteristics of 3-homogeneous permutation groups, it is proved that: there are no 6–(v, k, 2) designs D admitting a flag transitive group G ≤ Aut (D) of automorphisms.

We propose a novel framework for fast regularization of matrix-valued images. The resulting algorithms allow a unified treatment for a broad set of matrix groups and manifolds. Using an augmented-Lagrangian technique, we formulate a fast and highly parallel algorithm for matrix- valued image regularization. We demonstrate the applicability of the framework for various problems, such as motion analysis and diffusion tensor image reconstruction, show the formulation of the algorithm in terms of split- Bregman iterations and discuss the convergence properties of the proposed algorithms.

We consider finite nilpotent groups of matrices over commutative rings. A general result concerning the diagonalization of matrix groups in the terms of simple conditions for matrix entries is proven. We also give some arithmetic applications for representations over Dedekind rings.

We determine explicitly the Gauss sum τ l ( χ , e ) = ∑ X ∈ GL 2 ( Z / p l Z ) χ ( X ) e ( Tr X ) on the general linear group GL 2 ( Z / p l Z ) for every irreducible character χ of GL 2 ( Z / p l Z ) and a nontrivial additive character e of Z / p l Z , where p is an odd prime and l is an integer ⩾2. While there are several studies of the Gauss sums on finite algebraic groups defined over a finite field, this paper seems to be the first one which determines the Gauss sums on a matrix group over a finite ring.

In this paper, we consider a class of general coupled linear matrix equations over the complex number field. The mentioned coupled linear matrix equations contain the unknown complex matrix groups $$X=(X_1,X_2,\ldots ,X_q)$$ and $$Z=(Z_1,Z_2,\ldots ,Z_q)$$ . The conjugate and transpose of the unknown matrices $$X_i$$ and $$Z_i$$ , $$i\in I[1,q]$$ , appear in the considered coupled linear matrix equations. An iterative algorithm is presented to determine the unknown matrix groups $$X$$ and $$Z$$ such that $$X$$ and $$Z$$ are the groups of the generalized reflexive and anti-reflexive matrices, respectively. The proposed algorithm determines the solvability of the general coupled linear matrix equations over the generalized reflexive and anti-reflexive matrices, automatically. When the general coupled linear matrix equations are consistent over the generalized reflexive and anti-reflexive matrices, it is shown that the algorithm converges within finite number of steps, in the exact arithmetic. In addition, the optimal approximately generalized reflexive and anti-reflexive solution groups to the given arbitrary matrix groups $$\Gamma _x=(\Gamma _{1x},\Gamma _{2x},\ldots ,\Gamma _{qx})$$ and $$\Gamma _z=(\Gamma _{1z},\Gamma _{2z},\ldots ,\Gamma _{qz})$$ are derived. Finally, some numerical results are given to illustrate the validity of the presented theoretical results and feasibly of the proposed algorithm.

Classical group representation theory is used to determine which finite groups have finite negative K-theory. There follows a computation of the K −1 of integral group rings ZG for all finite non-abelian subgroups of the group SU(2) of unit quaternions. In principle, the method applies to any finite group.

In this paper, we show that the symmetric group can be characterized by its order and degree pattern. In fact, we get the following theorem: Let G be a finite group such that and . Then G is isomorphisic to one of the almost simple groups: and . Particularly, is 3-fold OD-characterizable.

By definition, admissible matrix groups are those that give rise to a wavelet-type inversion formula. This paper investigates necessary and sufficient admissibility conditions for abelian matrix groups. We start out by deriving a block diagonalization result for commuting real valued matrices. We then reduce the question of deciding admissibility to the subclass of connected and simply connected groups, and derive a general admissibility criterion for exponential solvable matrix groups. For abelian matrix groups with real spectra, this yields an easily checked necessary and sufficient characterization of admissibility. As an application, we sketch a procedure how to check admissibility of a matrix group generated by finitely many commuting matrices with positive spectra., Slightly revised; 27 pages

This article describes a setup that – given a composition tree – provides functionality for calculation in finite matrix groups using the Trivial-Fitting approach that has been used successfully for permutation groups. It treats the composition tree as a black-box object. It thus is applicable to other classes of groups for which a composition tree can be obtained. As an example, we consider an effective algorithm for determining conjugacy class representatives.

In this article, the invertible matrices over commutative semirings are studied. Some properties and equivalent descriptions of the invertible matrices are given and the inverse matrix of an invertible matrix is presented by analogues of the classic adjoint matrix. Also, Cramer's rule over a commutative semiring is established. The main results obtained in this article generalize the corresponding results for matrices over commutative rings, for lattice matrices, for incline matrices, for matrices over zerosumfree semirings and for matrices over additively regular semirings.

We define a new operation of multiplication on the set of square matrices. We determine when this multiplication is associative and when the set of matrices with this multiplication and the ordinary addition of matrices constitutes a ring. Furthermore, we determine when the nonstandard product admits the identity element and which elements are invertible. We study the relation between the nonstandard product and the affine transformations of a vector space. Using these results, we prove that the Mikhaĭlichenko group, which is a group of matrices with the nonstandard product, is isomorphic to a subgroup of matrices of a greater size with the ordinary product.

Many key invariants in the representation theory of classical groups (symmetric groups $S_n$, matrix groups $GL_n$, $O_n$, $Sp_{2n}$) are polynomials in $n$ (e.g., dimensions of irreducible representations). This allowed Deligne to extend the representation theory of these groups to complex values of the rank $n$. Namely, Deligne defined generically semisimple families of tensor categories parametrized by $n\in \mathbb{C}$, which at positive integer $n$ specialize to the classical representation categories. Using Deligne's work, Etingof proposed a similar extrapolation for many non-semisimple representation categories built on representation categories of classical groups, e.g., degenerate affine Hecke algebras (dAHA). It is expected that for generic $n\in \mathbb{C}$ such extrapolations behave as they do for large integer $n$ ("stabilization"). The goal of our work is to provide a technique to prove such statements. Namely, we develop an algebro-geometric framework to study categories indexed by a parameter $n$, in which the set of values of $n$ for which the category has a given property is constructible. This implies that if a property holds for integer $n$, it then holds for generic complex $n$. We use this to give a new proof that Deligne's categories are generically semisimple. We also apply this method to Etingof's extrapolations of dAHA, and prove that when $n$ is transcendental, "finite-dimensional" simple objects are quotients of certain standard induced objects, extrapolating Zelevinsky's classification of simple dAHA-modules for $n\in \mathbb{N}$. Finally, we obtain similar results for the extrapolations of categories associated to wreath products of the symmetric group with associative algebras., 26 pages; Comments welcome

We study the real elements in triangular matrix groups. We describe some classes of elements that are real in T n (K) – the groups of upper triangular matrices over a commutative field K. From the obtained results there follow some applications for finding real elements in general linear groups – GL n (K).

Firstly, we give conditions for the existence of the group inverse of the 2 × 2 matrixM=acbdover an arbitrary ring R with unity 1. Our second purpose is to derive a representation for the group inverse of M in the case when either the entry a or d has group inverse in R. We illustrate our results with some numerical examples in the setting of 2 × 2 complex block matrices.

In this article we consider the elements in upper triangular finite and infinite dimensional matrix groups over fields, whose order is equal to (). For the case when the characteristic of does not divide, we give a description of such elements. Next using this criterion, we show how to find the number of these elements in finite dimensional groups when is a finite field.

An elliptic curve E defined over a finite field K, E(K) is the set of solutions to the general Weierstrass polynomial E: y 2 + a1xy + a3y = x 3 + a2x 2 + a4x + a6 where the coefficients a1, a2, a3, a4, a6 є K. There exist a well defined addition of points on each curve such that the points form an abelian group under the addition operation. This group is either cyclic or isomorphic to the product of two cyclic groups. These set of solutions that form the group lie in the closure of the field K over which the curve is defined. If we allow the set to lie only in a particular extension of K, the addition operation is well defined there too. Therefore we can associate a group to every extension K' of the field K denoted by E(K'). Will the structure of the group defined over the base field K, be affected if the same group is made to lie in the extension K' of K?

Gaussian graphical models have become a well-recognized tool for the analysis of conditional independencies within a set of continuous random variables. From an inferential point of view, it is important to realize that they are composite exponential transformation families. We reveal this structure by explicitly describing, for any undirected graph, the (maximal) matrix group acting on the space of concentration matrices in the model. The continuous part of this group is captured by a poset naturally associated to the graph, while automorphisms of the graph account for the discrete part of the group. We compute the dimension of the space of orbits of this group on concentration matrices, in terms of the combinatorics of the graph; and for dimension zero we recover the characterization by Letac and Massam of models that are transformation families. Furthermore, we describe the maximal invariant of this group on the sample space, and we give a sharp lower bound on the sample size needed for the existence of equivariant estimators of the concentration matrix. Finally, we address the issue of robustness of these estimators by computing upper bounds on finite sample breakdown points., Published in at http://dx.doi.org/10.1214/13-AOS1130 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

I give my view of the early history of the discovery of hyperbolic structures on knot complements from my early work on representations of knot groups into matrix groups to my meeting with William Thurston in 1976. (This article was written by Robert Riley about ten years before his death in 2000 and never submitted for publication. An explanation of why it is being published now and some information about Riley and this article is given in the article by Brin, Jones and Singerman which accompanies this article in this issue of the [journal | arxiv].), 14 pages, 1 figure