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For positive integers u and v , let L u = [ 1 0 u 1 ] and R v = [ 1 v 0 1 ]. Let G u , v be the group generated by L u and R v. In a previous paper, the authors determined a characterization of matrices M = [ a b c d ] in G u , v when u , v ≥ 3 in terms of the short continued fraction representation of b / d. We extend this result to the case where u v ≥ 5. Additionally, we compute [ G u , v : G u , v ] where G u , v = { [ 1 + u v n 1 v n 2 u n 3 1 + u v n 4 ] ∈ SL 2 (Z) : (n 1 , n 2 , n 3 , n 4) ∈ Z 4 } for u , v ≥ 1 , extending a result of Chorna, Geller, and Shpilrain and solving the membership problem for G u , v with u , v ≥ 1. [ABSTRACT FROM AUTHOR]

For positive integers u and v, let Lu = [1 & 0 \\ u & 1] and Rv = [1 & v \\ 0 & 1 ]. Let Su,v be the monoid generated by Lu and Rv, and let Gu,v be the group generated by Lu and Rv. In this paper we expand on a characterization of matrices M = [a & b \\ c & d] in Sk,k and Gk,k when k ≥ 2 given by Esbelin and Gutan to Su,v when u,v ≥ 2 and Gu,v when u,v ≥ 3. We give a simple algorithmic way of determining if M is in Gu,v using a recursive function and the short continued fraction representation of b/d. [ABSTRACT FROM AUTHOR]

Studying the solvable subgroups of 2 × 2 matrix groups over Z, we find a maximal finite order primitive solvable subgroup of GL(2, Z) unique up to conjugacy in GL(2, Z). We describe the maximal primitive solvable subgroups whose maximal abelian normal divisor coincides with the group of units of a quadratic ring extension of Z. We prove that every real quadratic ring R determines h classes of conjugacy in GL(2, Z) of maximal primitive solvable subgroups of GL(2, Z), where h is the number of ideal classes in R. [ABSTRACT FROM AUTHOR]

We study matrix semigroups in which ring commutators have real spectra. We prove that irreducible semigroups with this property are simultaneously similar to semigroups of real-entried matrices. We also obtain a structure theorem for compact groups satisfying the property under investigation. [ABSTRACT FROM AUTHOR]

Reduction algorithms are an important tool for understanding structural properties of groups. They play an important role in algorithms designed to investigate matrix groups over a finite field. One such algorithm was designed by Brooksbank et al. for members of the class C6 in Aschbacher's theorem, namely groups N that are normalizers in GL(d; q) of certain absolutely irreducible symplectic-typeγ-groups R, whereγis a prime and d = rn with n > 2. However, the analysis of this algorithm has only been completed when d = r2 and when d = rn and n > 2, in the latter case under the condition that G=RZ(G)≃= N=RZ(N) . We prove that the algorithm runs successfully for some groups in the case of d = r3 without any assumption. [ABSTRACT FROM AUTHOR]

We study the abelian normal divisors of soluble matrix groups. [ABSTRACT FROM AUTHOR]

Our aim is to study topological minimality of some natural matrix groups. We show that the special upper triangular group S T + (n , F) is minimal for every local field F of characteristic ≠2. This result is new even for the field R of reals and it leads to some important consequences. We prove criteria for the minimality and total minimality of the special linear group SL (n , F) , where F is a subfield of a local field. This extends some known results of Remus–Stoyanov (1991) and Bader–Gelander (2017). One of our main applications is a characterization of Fermat primes, which asserts that for an odd prime p the following conditions are equivalent: (1) p is a Fermat prime; (2) SL (p − 1 , Q) is minimal, where Q is the field of rationals equipped with the p -adic topology; (3) SL (p − 1 , Q (i)) is minimal, where Q (i) ⊂ C is the Gaussian rational field. [ABSTRACT FROM AUTHOR]

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. We define three-sorted algebras with pseudomatrix multiplication, as well as physical structures, and establish their categorical equivalence. [ABSTRACT FROM AUTHOR]

Let R ∈ C n × n be a { k } -involutory matrix (that is, R k = I n ) for some integer k ≥ 2 , and let s be a nonnegative integer. A matrix A ∈ C n × n is called an { R , s + 1 , k } -potent matrix if A satisfies R A = A s + 1 R . In this paper, a matrix group corresponding to a fixed { R , s + 1 , k } -potent matrix is explicitly constructed, and properties of this group are derived and investigated. This group is then reconciled with the classical matrix group G A that is associated with a generalized group invertible matrix A . [ABSTRACT FROM AUTHOR]

Abstract: 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. [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]

We study soluble matrix groups over commutative rings and transfer some results of the theory of soluble matrix groups over a field to the case of soluble matrix groups over commutative rings. [ABSTRACT FROM AUTHOR]

Let F be a field, and let $\mathbb{G}$ be the standard Borel subgroup of the symplectic group Sp(2 m, F). In this paper, we characterize the bijective maps ϕ: $\mathbb{G}$ → $\mathbb{G}$ satisfying ϕ[ x, y] = [ ϕ( x), ϕ( y)]. [ABSTRACT FROM AUTHOR]

Abstract: We establish a correspondence between Young diagrams and differential operators of infinitely many variables. These operators form a commutative associative algebra isomorphic to the algebra of the conjugated classes of finite permutations of the set of natural numbers. The Schur functions form a complete system of common eigenfunctions of these differential operators, and their eigenvalues are expressed through the characters of symmetric groups. The structure constants of the algebra are expressed through the Hurwitz numbers. [Copyright &y& Elsevier]

Abstract: We present a new algorithm to decide finiteness of matrix groups defined over a field of positive characteristic. Together with previous work for groups in zero characteristic, this provides the first complete solution of the finiteness problem for finitely generated matrix groups over a field. We also give an algorithm to compute the order of a finite matrix group over a function field of positive characteristic by constructing an isomorphic copy of the group over a finite field. Our implementations of these algorithms are publicly available in Magma. [Copyright &y& Elsevier]

Abstract: We provide a new, practical algorithm for deciding finiteness of matrix groups over function fields of zero characteristic. The algorithm has been implemented in GAP. Experimental results and extensions of the algorithm to any field of zero characteristic are discussed. [Copyright &y& Elsevier]

Abstract: We develop methods for computing with matrix groups defined over a range of infinite domains, and apply those methods to the design of algorithms for nilpotent groups. In particular, we provide a practical nilpotency testing algorithm for matrix groups over an infinite field. We also provide algorithms to answer a number of structural questions for a nilpotent matrix group.The main algorithms have been implemented in GAP, for groups over the rational number field. [Copyright &y& Elsevier]

Abstract: Let be a regular covering projection of connected graphs with the group of covering transformations isomorphic to N. If N is an elementary abelian p-group, then the projection is called p-elementary abelian. The projection is vertex-transitive (edge-transitive) if some vertex-transitive (edge-transitive) subgroup of Aut lifts along , and semisymmetric if it is edge- but not vertex-transitive. The projection is minimal semisymmetric if cannot be written as a composition of two (nontrivial) regular covering projections, where is semisymmetric. Finding elementary abelian covering projections can be grasped combinatorially via a linear representation of automorphisms acting on the first homology group of the graph. The method essentially reduces to finding invariant subspaces of matrix groups over prime fields (see [A. Malnič, D. Marušič, P. Potočnik, Elementary abelian covers of graphs, J. Algebraic Combin. 20 (2004) 71–97]). In this paper, all pairwise nonisomorphic minimal semisymmetric elementary abelian regular covering projections of the Möbius–Kantor graph, the Generalized Petersen graph , are constructed. No such covers exist for . Otherwise, the number of such covering projections is equal to and in cases and , respectively, and to and in cases and , respectively. For each such covering projection the voltage rules generating the corresponding covers are displayed explicitly. [Copyright &y& Elsevier]

Abstract: We describe practical algorithms for computing a polycyclic presentation and for facilitating a membership test for a polycyclic subgroup of . A variation of this method can be used to check whether a finitely generated subgroup of is solvable or solvable-by-finite. We report on our implementations of the algorithms for determining a polycyclic presentation and checking solvability. [Copyright &y& Elsevier]

We study into the relationship between constructivizations of an associative commutative ring K with unity and constructivizations of matrix groups GLn (K) (general). SLn (K) (special). and UTn (K) (unitriangular) over K. It is proved that for n ≥ 3, a corresponding group is constructible if so is K. We also look at constructivizations of ordered groups. It turns out that a torsion-free constructible Abelian group is orderly constructible. It is stated that the unitriangular matrix group UTn (K) over an orderly constructible commutative associative ring K is itself orderly constructible. A similar statement holds also for finitely generated nilpotent groups, and countable free nilpotent groups. [ABSTRACT FROM AUTHOR]

Let G be a Lie group acting on a vector space V. I say that a vector field u:V→V is orbital if for all p in V, u(p) is tangent to the orbit of p at p. Now let the vector space V and the vector space tangent to G at the identity have inner products. In this setting I define a simple map (which I call quasi-projection) which transforms any vector field on V into an orbital one. I use the quasi-projection map to define flows which compute canonical forms. [Copyright &y& Elsevier]

The article discusses the constructive recognition of matrix groups from two tasks including decamping group into smaller components and dealing with irreducible cases. It states the use of Kung-Stong cycle index to compute generating function for proportion. It mentions the use of a Las Vegas algorithm to rewrite elements of classical group represented as irreducible subgroup.