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Let IA_n be the Torelli subgroup of Aut(F_n). We give an explicit finite set of generators for H_2(IA_n) as a GL_n(Z)-module. Corollaries include a version of surjective representation stability for H_2(IA_n), the vanishing of the GL_n(Z)-coinvariants of H_2(IA_n), and the vanishing of the second rational homology group of the level l congruence subgroup of Aut(F_n). Our generating set is derived from a new group presentation for IA_n which is infinite but which has a simple recursive form., Comment: 39 pages; minor revision; to appear in Geom. Topol

We compute the $L^2$-Betti numbers of the free $C^*$-tensor categories, which are the representation categories of the universal unitary quantum groups $A_u(F)$. We show that the $L^2$-Betti numbers of the dual of a compact quantum group $G$ are equal to the $L^2$-Betti numbers of the representation category $Rep(G)$ and thus, in particular, invariant under monoidal equivalence. As an application, we obtain several new computations of $L^2$-Betti numbers for discrete quantum groups, including the quantum permutation groups and the free wreath product groups. Finally, we obtain upper bounds for the first $L^2$-Betti number in terms of a generating set of a $C^*$-tensor category., v4: Minor changes, final version, to appear in Analysis and PDE. v3: We corrected a mistake in Section 3.2 without impact on the main results of the paper. v2: No major changes, but several remarks and a few references were added

In this paper, a novel algorithm “firework optimization” (FO) is introduced, as well as its hybridization with generating set search (GSS). FO is an exploration algorithm, which intends to explore a vast region within the search space in order to determine regions that may contain a regional optimum. As the regions with potential are defined, GSS, the exploitation algorithm is applied to determine whether regions determined by FO contain a local or a global optimum. The validity of FO and GSS hybridization algorithm is verified using Branin and Six-Hump Camel test functions, and the hybridization algorithm is applied to an interior permanent magnet synchronous motor, in order to minimize torque ripple.

We define a new invariant of a conjugacy class of subgroups which we call the breadth and prove that a quasiconvex subgroup of a negatively curved group has finite breadth in the ambient group. Utilizing the coset graph and the geodesic core of a subgroup we give an explicit algorithm for constructing a finite generating set for an intersection of a quasiconvex subgroup of a negatively curved group with its conjugate. Using that algorithm we construct algorithms for computing the breadth, the width, and the height of a quasiconvex subgroup of a negatively curved group. These algorithms decide if a quasiconvex subgroup of a negatively curved group is almost malnormal in the ambient group. We also explicitly compute a quasiconvexity constant of the intersection of two quasiconvex subgroups and give examples demonstrating that height, width, and breadth are different invariants of a subgroup.

In this note, we introduce the notions of color-permutable automorphisms and color-permutable vertex-transitive Cayley graphs of semigroups. As a main result, for a finite monoid S and a generating set C of S, we explicitly determine the color-permutable automorphism group of \(\mathrm {Cay}(S,C)\) [Theorem 1.1]. Also for a monoid S and a generating set C of S, we explicitly determine the color-preserving automorphism group (endomorphism monoid) of \(\mathrm {Cay}(S,C)\) [Proposition 2.3 and Corollary 2.4]. Then we use these results to characterize when \(\mathrm {Cay}(S,C)\) is color-endomorphism vertex-transitive [Theorem 3.4].

Let r i be positive integers and R i = Z 2 [ x ] / 〈 x r i − 1 〉 for 1 ≤ i ≤ l . Denote R = R 1 × R 2 × ⋯ × R l . Generalized quasi-cyclic (GQC) code C of length ( r 1 , r 2 , … , r l ) over Z 2 can be viewed as Z 2 [ x ] -submodule of R . In this paper, we investigate the algebraic structure of C by presenting its normalized generating set. We also present a method to determine the normalized generating set of the dual code of C , which is derived from the normalized generating set of C .

A finite lattice is interval dismantlable if it can be partitioned into an ideal and a filter, each of which can be partitioned into an ideal and a filter, etc., until you reach 1-element lattices. In this note, we find a quasi-equational basis for the pseudoquasivariety of interval dismantlable lattices, and show that there are infinitely many minimal interval non-dismantlable lattices.

Consider a finite dimensional vector space V over a finite field F q . We give a minimal generating set for the ring of invariants F q [ V ⊕ V ⁎ ] GL ( V ) , and show that this ring is a Gorenstein ring but is not a complete intersection. These results confirm a conjecture of Bonnafe and Kemper [3, Conjecture 3.1] .

We consider groups [Formula: see text] which have generating sets consisting of pairwise conjugate elements. We introduce a natural condition on such sets, namely path-connectedness, which has strong consequences when [Formula: see text] acts on a real tree. The main result of the paper states that the group of type-preserving automorphisms of a regular simplicial tree has a generating set with this property.

The standard (n, k, d) model of random groups is a model where the relators are chosen randomly from the set of cyclically reduced words of length k on an n-element generating set. Gromov’s density model of random groups considers the case where n is fixed, and k tends to infinity. We instead fix k, and let n tend to infinity. We prove that for all k ≥ 2 at density d > 1/2 a random group in this model is trivial or cyclic of order two, whilst for d < 1 such 2 a random group is infinite and hyperbolic. In addition we show that for d < 1/k such a random k group is free, and that this threshold is sharp. These extend known results for the triangular (k = 3) and square (k = 4) models of random groups. Postprint

Powers of (monomial) ideals is a subject that still calls attraction in various ways. Let $I\subset \mathbb K[x_1,\ldots,x_n]$ be a monomial ideal and let $G(I)$ denote the (unique) minimal monomial generating set of $I$. How small can $|G(I^i)|$ be in terms of $|G(I)|$? We expect that the inequality $|G(I^2)|>|G(I)|$ should hold and that $|G(I^i)|$, $i\ge 2$, grows further whenever $|G(I)|\ge 2$. In this paper we will disprove this expectation and show that for any $n$ and $d$ there is an $\mathfrak m$-primary monomial ideal $I\subset \mathbb K[x_1,\ldots,x_n]$ such that $|G(I)|>|G(I^i)|$ for all $i\le d$., 9 pages, 3 figures

In the first part of this paper, we discuss the classical W-algebra $\mathcal{W}(\mathfrak{g}, F)$ associated with a Lie superalgebra $\mathfrak{g}$ and the nilpotent element $F$ in an $\mathfrak{sl}_2$-triple. We find a generating set of $\mathcal{W}(\mathfrak{g}, F)$ and compute the Poisson brackets between them. In the second part, which is the main part of the paper, we discuss supersymmetric classical W-algebras. We introduce two different constructions of a supersymmetric classical W-algebra $\mathcal{W}(\mathfrak{g}, f)$ associated with a Lie superalgebra $\mathfrak{g}$ and an odd nilpotent element $f$ in a subalgebra isomorphic to $\mathfrak{osp}(1|2)$. The first construction is via the SUSY classical BRST complex and the second is via the SUSY Drinfeld-Sokolov Hamiltonian reduction. We show that these two methods give rise to isomorphic SUSY Poisson vertex algebras. As a supersymmetric analogue of the first part, we compute explicit generators and Poisson brackets between the generators.

The canonical ideal for Harbater Katz Gabber covers satisfying the conditions of Petri's theorem is studied and an explicit non-singular model of the above curves is given., Comment: 17 pages

A zig-zag order is like a directed path, only with alternating directions. A generating set of minimal size for the semigroup of all full transformations on a finite set preserving the zig-zag order was determined by Fenandes et al. in 2019. This paper deals with generating sets of the semigroup $F_{\mathbb{N}}$ of all full transformations on the set of all natural numbers preserving the zig-zag order. We prove that $F_{\mathbb{N}}$ has no minimal generating sets and present two particular infinite decreasing chains of generating sets of $F_{\mathbb{N}}$., Comment: 17 pages

Invariable generation is a topic that has predominantly been studied for finite groups. In 2014, Kantor, Lubotzky, and Shalev produced extensive tools for investigating invariable generation for infinite groups. Since their paper, various authors have investigated the property for particular infinite groups or families of infinite groups. A group is invariably generated by a subset $S$ if replacing each element of $S$ with any of its conjugates still results in a generating set for $G$. In this paper we investigate how this property behaves with respect to wreath products. Our main work is to deal with the case where the base of $G\wr_X H$ is not invariably generated. We see both positive and negative results here depending on $H$ and its action on $X$., Comment: 11 pages; to appear in J. Group Theory

A subset \(X\) of prime power order elements of a finite group \(G\) is called pp-independent if there is no proper subset \(Y\) of \(X\) such that \(\langle Y,\Phi(G) \rangle = \langle X,\Phi(G) \rangle\), where \(\Phi(G)\) is the Frattini subgroup of \(G\). A group \(G\) has property \(\mathcal{B}_{pp}\) if all pp-independent generating sets of \(G\) have the same size. \(G\) has the pp-basis exchange property if for any pp-independent generating sets \(B_1, B_2\) of \(G\) and \(x\in B_1\) there exists \(y\in B_2\) such that \((B_1\setminus \{x\})\cup \{y\}\) is a pp-independent generating set of \(G\). In this paper we describe all finite solvable groups with property \(\mathcal{B}_{pp}\) and all finite solvable groups with the pp-basis exchange property.

We derive the polynomial representations for minimal relations of the generating set of numerical semigroups \(R_n^k=\langle (n-1)^k,n^k,(n+1)^k\rangle \), \(k=2,3\). We find also the polynomial representations for degrees of syzygies in the Hilbert series \(H\left( z,R_n^k\right) \) of these semigroups, their Frobenius numbers \(F\left( R_n^k\right) \) and genera \(G\left( R_n^k\right) \). We discuss an extension of polynomial representations for minimal relations on numerical semigroups \(R_n^k\), \(k\ge 4\).

We show that any subgroup of a finitely generated virtually abelian group $G$ grows rationally relative to $G$, that the set of right cosets of any subgroup of $G$ grows rationally, and that the set of conjugacy classes of $G$ grows rationally. These results hold regardless of the choice of finite weighted generating set for $G$., Comment: 36 pages. Updated according to referee's suggestions. To appear in Illinois Journal of Mathematics

Let Z^+denotes the set of all positive integers. Let X_n={1,2,…,n} be the finite chain for n∈Z^+ and let T_n be the full transformation semigroup on X_n. Also let 〖OCT〗_n and 〖ORCT〗_n be the semigroup of order-preserving full contraction mappings, and the semigroup of order-preserving or order-reversing full contraction mappings on X_n, respectively. It is well-known that 〖OCT〗_n and 〖ORCT〗_n are subsemigroups of T_n. In this paper we obtain ranks of the semigroups 〖OCT〗_n and 〖ORCT〗_n., Z^+, tüm pozitif tamsayıların kümesi olsun. n∈Z^+ için X_n={1,2,…,n} sonlu bir zincir ve T_n, X_n üzerindeki tam dönüşümler yarıgrubu olsun. Ayrıca 〖OCT〗_n ve 〖ORCT〗_n sırasıyla X_n üzerindeki sıra-koruyan tam daralma dönüşümler yarıgrubu ve sıra-koruyan veya sıra-çeviren tam daralma dönüşüler yarıgrubu olsun.〖OCT〗_n ve 〖ORCT〗_n yarıgruplarının T_n yarıgrubunun altyarıgrupları olduğu bilinmektedir. Bu çalışmada 〖OCT〗_n ve 〖ORCT〗_n yarıgruplarının rankları araştırılmıştır.

A zig-zag (or fence) order is a special partial order on a (finite) set. In this paper, we consider the semigroup $$\mathscr {TF}_{n}$$ of all order-preserving transformations on an n-element zig-zag-ordered set. We determine the rank of $$\mathscr {TF}_{n}$$ and provide a minimal generating set for $$ \mathscr {TF}_{n}$$ . Moreover, a formula for the number of idempotents in $$\mathscr {TF}_{n}$$ is given.

In order to meet the functional requirements of variable speed diesel generators and inverters, this paper proposes a speed regulation, rectification and inverter scheme. And the overall scheme is designed. Then the modelling, simulation and hardware design are carried out. Through the digital intelligent control system, the diesel generator set can change its speed according to the magnitude of the electric load, which can reduce the consumption of oil, save energy, and reduce the loss. Firstly, this paper analyzes the function requirements of diesel generator and inverter and lays out the scheme of adjusting speed, the rectifier and the inverter. Besides, the paper designs the topology structure for inverter. According to the overall design program, the system of inverter is modeled and simulated in this paper. And the strategy of booster and inverter is determined. Secondly, the design of hardware for inverter system is finished, and the power and control boards are developed. What’s more, the units of hardware are debugged. With the development of hardware, this paper introduces a program which can adjust the speed according to the change of load. In the end, DC bus voltage regulator can be controlled in different combinations of speed and load, which can output the power reaching the quality requirements by SPWM’s control.

We study the hydrodynamic-type system of differential equations modeling isothermal no-slip drift flux. Using the facts that the system is partially coupled and its subsystem reduces to the (1+1)-dimensional Klein--Gordon equation, we exhaustively describe generalized symmetries, cosymmetries and local conservation laws of this system. A generating set of local conservation laws under the action of generalized symmetries is proved to consist of two zeroth-order conservation laws. The subspace of translation-invariant conservation laws is singled out from the entire space of local conservation laws. We also find broad families of local recursion operators and a nonlocal recursion operator, and construct an infinite family of Hamiltonian structures involving an arbitrary function of a single argument. For each of the constructed Hamiltonian operators, we obtain the associated algebra of Hamiltonian symmetries., 36 pages, extended version, the proof on cosymmetries in presented with more details

In this paper we provide a comparative empirical analysis of four different generating sets for the algebraic Differential Evolution for Permutations (DEP) applied to the Traveling Salesman Problem (TSP). In particular, DEP has been extended in order to use the reversal moves as generating set. Two different randomized decomposers are proposed for the reversal generators. The experiments have been conducted on a selected set of commonly adopted TSP instances, and the results show the newly proposed generating set leads to better performances with respect to other three generating sets based on alternative search moves.

There is an abnormal noise generated from the timing gears while practicing for the commissioning of a diesel generator set. As the load increases, this phenomenon becomes more apparent. In order to find the cause of the noise, a fault diagnosis model is established which considers not only the gears but also the effects of the associated structures. After eliminating the possible reasons such as the defects of timing gears, the abnormality of the valve system, fuel system controllers and generator, the injector problem is suspected to cause the fault. To verify this conjecture, the fuel injection pumps of the diesel generator set are inspected and the second injector is found to be blackened with some carbon deposition. By replacing the injector, the noise is gone. Hence, the fault diagnosis model can provide a new tool for future diagnosis of gear fault.

We consider a transitive action of a finitely generated group $G$ and the Schreier graph $\Gamma$ defined by this action for some fixed generating set. For a probability measure $\mu$ on $G$ with a finite first moment we show that if the induced random walk is transient, it converges towards the space of ends of $\Gamma$. As a corollary we obtain that for a probability measure with a finite first moment on Thompson's group $F$, the support of which generates $F$ as a semigroup, the induced random walk on the dyadic numbers has a non-trivial Poisson boundary. Some assumption on the moment of the measure is necessary as follows from an example by Juschenko and Zheng., Comment: 10 pages, 2 figures. Changes from previous version: Added Proposition 2.6. Updated references

We develop a practical algorithm to decide whether a finitely generated subgroup of a solvable algebraic group $G$ is arithmetic. This incorporates a procedure to compute a generating set of an arithmetic subgroup of $G$. We also provide a simple new algorithm for integrality testing of finitely generated solvable-by-finite linear groups over the rational field. The algorithms have been implemented in {\sc Magma}.

For power generation purposes, wound-field synchronous generators are often preferred to permanent magnet machines, due to the need of controlling the field winding voltage and current for guaranteeing a constant voltage at the generator's armature terminals. In a traditional generating set, this function is accomplished by a relatively complex feedback control system which typically comprises an excitation system and an automatic voltage regulator. Due to their excellent and proven performance capability, such classical generating sets have seen only strictly incremental improvements in the last 60 years or so. However, today, there is an interest in revamping their design and development, partly due to the even increasing efficiency and reliability requirements and partly due to advances in material and manufacturing techniques. This work investigates the feasibility of a hybrid permanent magnet - field winding excitation for synchronous generators' exciters. This design solution aims at reducing the losses of the exciter and thus of the whole generating system, while also increasing its reliability.

Let $G$ be an infinite group and let $X$ be a finite generating set for $G$ such that the growth series of $G$ with respect to $X$ is a rational function; in this case $G$ is said to have rational growth with respect to $X$. In this paper a result on sizes of spheres (or balls) in the Cayley graph $\Gamma(G,X)$ is obtained: namely, the size of the sphere of radius $n$ is bounded above and below by positive constant multiples of $n^\alpha \lambda^n$ for some integer $\alpha \geq 0$ and some $\lambda \geq 1$. As an application of this result, a calculation of degree of commutativity (d. c.) is provided: for a finite group $F$, its d. c. is defined as the probability that two randomly chosen elements in $F$ commute, and Antol\'in, Martino and Ventura have recently generalised this concept to all finitely generated groups. It has been conjectured that the d. c. of a group $G$ of exponential growth is zero. This paper verifies the conjecture (for certain generating sets) when $G$ is a right-angled Artin group or, more generally, a graph product of groups of rational growth in which centralisers of non-trivial elements are "uniformly small"., Comment: 16 pages; final version

Let $G$ be a group and let $S$ be an inverse-closed and identity-free generating set of $G$. The \emph{Cayley graph} $\Cay(G,S)$ has vertex-set $G$ and two vertices $u$ and $v$ are adjacent if and only if $uv^{-1}\in S$. Let $CAY_d(n)$ be the number of isomorphism classes of $d$-valent Cayley graphs of order at most $n$. We show that $\log(CAY_d(n))\in\Theta (d(\log n)^2)$, as $n\to\infty$. We also obtain some stronger results in the case $d=3$., Comment: 15 pages

We characterise the elements of the (maximum) idempotent generated subsemigroup of the Kauffman monoid in terms of combinatorial data associated to certain normal forms. We also calculate the smallest size of a generating set and idempotent generating set., Comment: 10 pages, 1 figure; v2 incorporates referee's suggestions (to appear in Glasgow Math J)

For every metric space $\mathcal X$ in which there exists a sequence of finite groups of bounded-size generating set that does not embed coarsely, and for every unbounded, increasing function $\rho$, we produce a group of subexponential word growth all of whose imbeddings in $\mathcal X$ have distortion worse than $\rho$. This applies in particular to any B-convex Banach space $\mathcal X$, such as Hilbert space.

Let K be a field of characteristic char ( K ) ≠ 2 , 3 and let E be an elliptic curve defined over K. Let m be a positive integer, prime with char ( K ) if char ( K ) ≠ 0 ; we denote by E [ m ] the m-torsion subgroup of E and by K m : = K ( E [ m ] ) the field obtained by adding to K the coordinates of the points of E [ m ] . Let P i : = ( x i , y i ) ( i = 1 , 2 ) be a Z -basis for E [ m ] ; then K m = K ( x 1 , y 1 , x 2 , y 2 ) . We look for small sets of generators for K m inside { x 1 , y 1 , x 2 , y 2 , ζ m } trying to emphasize the role of ζ m (a primitive m-th root of unity). In particular, we prove that K m = K ( x 1 , ζ m , y 2 ) , for any odd m ⩾ 5 . When m = p is prime and K is a number field we prove that the generating set { x 1 , ζ p , y 2 } is often minimal, while when the classical Galois representation Gal ( K p / K ) → GL 2 ( Z / p Z ) is not surjective we are sometimes able to further reduce the set of generators. We also describe explicit generators, degree and Galois groups of the extensions K m / K for m = 3 and m = 4 .

For a field K, a square-free monomial ideal I of K[x1,…,xn] is called an f-ideal if both its facet complex and Stanley-Reisner complex have the same f-vector. In this paper, we introduce a combinatorial concept (LU-set) and use it to characterize an (n,d)th f-ideal, whose minimal monomial generating set consists of some monomials of a common degree d from K[x1,…,xn]. We classify all (n,2)th f-ideals, thus list all f-graphs whose edge ideals are exactly the (n,2)th f-ideals. Furthermore, we show that all f-graphs are Cohen-Macaulay.

In this paper, we consider two regularized transformation-optics cloaking schemes for electromagnetic (EM) waves. Both schemes are based on the blowup construction with the generating sets being, respectively, a generic curve and a planar subset. We derive sharp asymptotic estimates in assessing the cloaking performances of the two constructions in terms of the regularization parameters and the geometries of the cloaking devices. The first construction yields an approximate full-cloak, whereas the second construction yields an approximate partial-cloak. Moreover, by incorporating properly chosen conducting layers, both cloaking constructions are capable of nearly cloaking arbitrary EM contents. This work complements the existing results in [5--7] on approximate EM cloaks with the generating set being a singular point, and it also extends [9,25] on regularized full and partial cloaks for acoustic waves governed by the Helmholtz system to the more challenging EM case governed by the full Maxwell system., Comment: 34 pages, 2 figures

We study cubic graphical regular representations of the finite simple groups PSL 2 ( q ) . It is shown that such graphical regular representations exist if and only if q ? 7 , and the generating set must consist of three involutions.

For a right vector space we study the question whether any generating set of subspaces of a bounded finite dimension contains a generating subset minimal with respect to inclusion. We obtain partial positive results as well as a complete answer to the set theoretic analogy of the question.

In our preceding paper a generating set of the derived Picard group of a selfinjective Nakayama algebra was constructed combining some previous results for Brauer tree algebras and the technique of orbit categories developed there. In this paper we finish the computation of the derived Picard group of a selfinjective Nakayama algebra.

Let M n ( F ) be the algebra of n × n matrices and let S be a generating set of M n ( F ) as an F -algebra. The length of a finite generating set S of M n ( F ) is the smallest number k such that words of length not greater than k generate M n ( F ) as a vector space. Traditionally the identity matrix is assumed to be automatically included in all generating sets S and counted as a word of length 0. In this paper we discuss how the problem changes if this assumption is removed.

We prove that an operator system is (min, ess)-nuclear if its C*-envelope is nuclear. This allows us to deduce that an operator system associated to a generating set of countable discrete group by Farenick et al. is (min, ess)-nuclear if and only if the group is amenable. We also make a detailed comparison between ess and other operator system tensor products and show that an operator system associated to a minimal generating set of a finitely generated discrete group (resp., a finite graph) is (min, max)-nuclear if and only if the group is of order less than or equal to 3 (resp., every component of the graph is complete)., Comment: To appear in J. Aust. Math. Soc

Let $I$ be an arbitrary ideal generated by binomials. We show that certain equivalence classes of fibers are associated to any minimal binomial generating set of $I$. We provide a simple and efficient algorithm to compute the indispensable binomials of a binomial ideal from a given generating set of binomials and an algorithm to detect whether a binomial ideal is generated by indispensable binomials., Comment: 15 pages, 2 figures. All results have been extended to the general case of binomial ideals. Several proofs were added/modified and there is a new theorem, Theorem 1.8. The paper will appear in Journal of Symbolic Computation

Based on Jeffcott rotor model, the coupled bending-torsional rotor-bearing system with rub-impact under electromagnetic excitation for hydraulic generating set is established and the corresponding equation is given. The influence of excitation current, mass eccentricity and electromagnetic torque in the system is investigated by taking use of numerical method. The simulation results show that eccentricity is the crucial factor causing the changes of system dynamic characteristics, compared to the effect of torsional degree of freedom. And the larger the eccentricity is, the more obviously it affects the system responses. While the complicated dynamic behavior of bending vibration response can be restrained, due to the introduction of torsion motion, which has the property to suppress complex dynamic response of system, particularly, the inhibited and improved effect is more evident, as the eccentricity turns to be smaller. In addition, the unstable jumping phenomena of torsional response can be motivated by electromagnetic torque, which has to be taken seriously during the process of model establishment and dynamic analysis of system. From the global view of electromechanical coupling characteristics, the coupled bending-torsional vibration model with rub-impact under electromagnetic excitation can supply a more comprehensive reflection about the dynamic characteristics of system and provide useful reference with system state recognition and fault diagnosis.

In this paper, we investigate the isoperimetric constant (or expansion constant) of a Paley graph, and the Kazhdan constant of the group and generating set associated with a Paley graph. ¶ We give two new upper bounds for the isoperimetric constant [math] for the Paley graph [math] . These bounds improve previously known eigenvalue bounds on [math] . Along with a known eigenvalue lower bound for [math] , they provide a narrow strip in which [math] must live. More precisely, we show that [math] , which implies that [math] . ¶ In addition, we show that the Kazhdan constant associated with the integers modulo [math] and the generating set for the Paley graph [math] approaches [math] as [math] tends to infinity, which is the best possible limit that the Kazhdan constant can be.

In this paper, first we investigate some properties of L-ordered groups such as L-subgroup generated by an arbitrary subset and product of L-ordered groups. Then we introduce the notions of totally L-ordered group and strong totally L-ordered group, where L is a frame and some related results are obtained. We show that the set of all isomorphisms on a totally L- subgroup, forms an L-ordered group. Moreover, if it is a strong totally L-ordered group, then the set of all isomorphisms on it is an L-lattice ordered group.

We find some sufficient conditions under which the permutational wreath product of two groups has a minimal (irredundant) generating set. In particular we prove that for a regular rooted tree the group of all automorphisms and the group of all finite-state automorphisms of such a tree satisfy these conditions. Thereby we solve the problem that was stated by B. Csakany and F. Gecseg in 1965.

We study p-adic integral models of certain PEL Shimura varieties with level subgroup at p related to the $${\Gamma_1(p)}$$ -level subgroup in the case of modular curves. We will consider two cases: the case of Shimura varieties associated with unitary groups that split over an unramified extension of $${\mathbb{Q}_p}$$ and the case of Siegel modular varieties. We construct local models, i.e. simpler schemes which are etale locally isomorphic to the integral models. Our integral models are defined by a moduli scheme using the notion of an Oort–Tate generator of a group scheme. We use these local models to find a resolution of the integral model in the case of the Siegel modular variety of genus 2. The resolution is regular with special fiber a nonreduced divisor with normal crossings.

In this paper we exhibit two infinite families of trees $\{T^1_n\}_{n \geq 17}$ and $\{T^2_n\}_{n \geq 17}$ on $n$ vertices, such that $T^1_n$ and $T^2_n$ are non-isomorphic, co-spectral, and the right-angled Coxeter groups (RACGs) based on $T^1_n$ and $T^2_n$ have the same geodesic growth with respect to the standard generating set. We then show that the spectrum of a tree does is not sufficient to determine the geodesic growth of the RACG based on that tree, by providing two infinite families of trees $\{S^1_n\}_{n \geq 11}$ and $\{S^2_n\}_{n \geq 11}$, on $n$ vertices, such that $S^1_n$ and $S^2_n$ are non-isomorphic, co-spectral, and the right-angled Coxeter groups (RACGs) based on $S^1_n$ and $S^2_n$ have distinct geodesic growth. Asymptotically, as $n\rightarrow \infty$, each set $T^i_n$, or $S^i_n$, $i=1,2$, has the cardinality of the set of all trees on $n$ vertices. Our proofs are constructive and use two families of trees previously studied by B. McKay and C. Godsil., Comment: 14 pages, 4 figures, a typo in formula (3) corrected; supplementary material and a SAGE worksheet available at http://sashakolpakov.wordpress.com/list-of-papers/

An involution r in a Coxeter group W is called an intrinsic reflection of W if r ∈ S W for each Coxeter generating set S of W. In recent joint work with R.B. Howlett [13] we determined all intrinsic reflections in finitely generated Coxeter groups. In the present paper we extend this result to the infinite rank case. An important tool in [13] is the notion of the finite continuation of an involution that is only meaningful for finitely generated Coxeter groups. Here we introduce the locally finite continuation for any subset of an arbitrary group which enables us to deal with Coxeter groups of infinite rank. We apply our result to show that certain classes of Coxeter groups are reflection independent and we investigate rigidity of 2-spherical Coxeter systems of arbitrary ranks.

A group [Formula: see text] has the Basis Property if every subgroup [Formula: see text] of [Formula: see text] has an equivalent basis (minimal generating set). We studied a special case of the finite group with the Basis Property, when [Formula: see text]-group [Formula: see text] is an abelian group. We found the necessary and sufficient conditions on an abelian [Formula: see text]-group [Formula: see text] of [Formula: see text] with the Basis Property to be kernel of Frobenius group.