Properties Of Such Groups Research Articles

The Structure of p-fixed Autonilpotent Finite Groups

The concept of autonilpotent groups was introduced by Moghaddam and Parvaneh in 2010 which is related to concepts of absolute centre and autocommutator subgroups that were first proposed by Hegarty in 1994 and 1997 respectively. In the present article we give some useful properties of such groups. In the first section, we define a subgroup GK_n for a given group G, as GK_n=〈 [g ,α^n ] ┤| g∈ G,α∈Aut(G) 〉 which is characteristic subgroup of G . In this section, based on the induction method, we obtain the structure of GK_n as 〈 [ g ,α ]^n ┤|g∈ G,α∈ Aut(G) 〉, 〈 [ g ,α ]^n [g,α,α]^(n(n-1)/2) ┤|g∈ G,α∈ Aut(G) 〉 and 〈 (∏_(i=0)^(n-1)▒〖[g,α] [g,α,α]^i) [g,α,α,α]^((n(n-1)(n-2))/6) 〗┤|g∈ G,α∈ Aut(G) 〉 in autonilpotent groups of classes 2, 3 and 4 respectively. In second section, n-fixed group is introduced as a group G with GK_n=1 for some n∈N. Then among other results of n-fixed groups, we characterize some n-fixed groups. Based on the role of the absolute centre subgroup in the structure of the group, we prove a non-trivial finite group G is isomorphic to Z_2 iff G is 1-fixed group also we show that abelian group G is p -fixed autonilpotent group iff it is isomorphic to G≅ C_(2^k ) where 1≤ k≤ 3

Ulam stability of lamplighters and Thompson groups

We show that a large family of groups is uniformly stable relative to unitary groups equipped with submultiplicative norms, such as the operator, Frobenius, and Schatten p-norms. These include lamplighters Γ≀Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma \\wr \\Lambda $$\\end{document} where Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document} is infinite and amenable, as well as several groups of dynamical origin such as the classical Thompson groups F,F′,T\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F, F', T$$\\end{document} and V. We prove this by means of vanishing results in asymptotic cohomology, a theory introduced by the second author, Glebsky, Lubotzky and Monod, which is suitable for studying uniform stability. Along the way, we prove some foundational results in asymptotic cohomology, and use them to prove some hereditary features of Ulam stability. We further discuss metric approximation properties of such groups, taking values in unitary or symmetric groups.

On the bottom layer in groups

The question of the possibility of restoring information about a group by its lower layer, that is, by the set of its elements of prime orders, is considered. The question is classical for mathematical modeling: restoration of the missing information about the object from the part of the preserved data. A group is said to be recognizable from the bottom layer under additional conditions if it is uniquely reconstructed from the bottom layer under these conditions. A group G is said to be almost recognizable from the bottom layer under additional conditions if there exists a finite number of pairwise nonisomorphic groups satisfying these conditions, with the same bottom layer as the group G. A group G is called unrecognizable from the bottom layer under additional conditions if there is an infinite the number of pairwise non-isomorphic groups that satisfy these conditions and have the same bottom layer as the group G. Results are given on the recognition of groups by the bottom layer in various classes of groups. The concept of recognizability by the lower layer was introduced by analogy with the actively studied recognizability by spectrum, that is, by the set of orders of group elements. In this paper, we consider groups that, without a single element, coincide with their bottom layer. Examples of groups with these conditions in the classes of Abelian and non-Abelian groups are given. The properties of such groups are established. The results can be applied when encoding information in space communications.

О периодических группах, насыщенных конечными группами Фробениуса

A group is called weakly conjugate biprimitively finite if each its element of prime order generates a finite subgroup with any of its conjugate elements. A binary finite group is a periodic group in which any two elements generate a finite subgroup. If $\mathfrak{X}$ is some set of finite groups, then the group $G$ saturated with groups from the set $\mathfrak{X}$ if any finite subgroup of $G$ is contained in a subgroup of $G$, isomorphic to some group from $\mathfrak{X}$. A group $G = F \leftthreetimes H$ is a Frobenius group with kernel $F$ and a complement $H$ if $H \cap H^f = 1$ for all $f \in F^{\#}$ and each element from $G \setminus F$ belongs to a one conjugated to $H$ subgroup of $G$. In the paper we prove that a saturated with finite Frobenius groups periodic weakly conjugate biprimitive finite group with a nontrivial locally finite radical is a Frobenius group. A number of properties of such groups and their quotient groups by a locally finite radical are found. A similar result was obtained for binary finite groups with the indicated conditions. Examples of periodic non locally finite groups with the properties above are given, and a number of questions on combinatorial group theory are raised.

On the categorical behaviour of V-groups

We consider compatible group structures on a V -category, where V is a quantale, and we study the topological and algebraic properties of such groups. Examples of such structures are preordered groups, metric and ultrametric groups, probabilistic (ultra)metric groups. In particular, we show that, when V is a frame, symmetric V -groups satisfy very strong categorical-algebraic properties, typical of the category of groups. In particular, symmetric V -groups form a protomodular category.

Emergence of social cluster by collective pairwise encounters in Drosophila.

Many animals exhibit an astonishing ability to form groups of large numbers of individuals. The dynamic properties of such groups have been the subject of intensive investigation. The actual grouping processes and underlying neural mechanisms, however, remain elusive. Here, we established a social clustering paradigm in Drosophila to investigate the principles governing social group formation. Fruit flies spontaneously assembled into a stable cluster mimicking a distributed network. Social clustering was exhibited as a highly dynamic process including all individuals, which participated in stochastic pair-wise encounters mediated by appendage touches. Depriving sensory inputs resulted in abnormal encounter responses and a high failure rate of cluster formation. Furthermore, the social distance of the emergent network was regulated by ppk-specific neurons, which were activated by contact-dependent social grouping. Taken together, these findings revealed the development of an orderly social structure from initially unorganised individuals via collective actions.

Про скінченні p-групи з недедекіндовою нормою абелевих нециклічних підгруп.

In this paper the finite p-groups, in which the norm of Abelian non-cyclic subgroups is Non-Dedekind one, are considered. There were defined the series of general properties of such groups. It was proved that the finite p-group, in which the norm of Abelian non-cyclic subgroups is non-Dedekind and doesn't contain the Abelian non-cyclic subgroups, also doesn't contain such subgroups.

Finite groups with a trivial Chermak–Delgado subgroup

AbstractThe Chermak–Delgado lattice of a finite group is a modular, self-dual sublattice of the lattice of subgroups ofG. The least element of the Chermak–Delgado lattice ofGis known as the Chermak–Delgado subgroup ofG. This paper concerns groups with a trivial Chermak–Delgado subgroup. We prove that if the Chermak–Delgado lattice of such a group is lattice isomorphic to a Cartesian product of lattices, then the group splits as a direct product, with the Chermak–Delgado lattice of each direct factor being lattice isomorphic to one of the lattices in the Cartesian product. We establish many properties of such groups and properties of subgroups in the Chermak–Delgado lattice. We define a CD-minimal group to be an indecomposable group with a trivial Chermak–Delgado subgroup. We establish lattice theoretic properties of Chermak–Delgado lattices of CD-minimal groups. We prove an extension theorem for CD-minimal groups, and use the theorem to produce twelve examples of CD-minimal groups, each having different CD lattices. Curiously, quasi-antichainp-group lattices play a major role in the author’s constructions.

Luminescent and electrical properties of oxygen‐implanted silicon

Light emitting diodes with an active defect‐rich region produced by oxygen implantation and a subsequent multistep annealing of silicon wafers were investigated by means of transmission electron microscopy, SIMS, capacitance voltage, deep level transient spectroscopy, electroluminescence (EL), and cathodoluminescence (CL) techniques. The properties of two groups of n‐based samples with and without thermal pre‐treatment at 1000 °C for 15 min were compared regarding their defect structure, defects electrical activity, luminescent spectra as well as the impact of prolonged intense electron irradiation. The observed difference in the properties of such groups was explained by a difference in the density and oxygen content of oxygen‐related defects. A significant distinction between EL and CL spectra at low excitation levels was found and interpreted to be due to particular defect kinds in near‐surface and the deepest layers of the implanted region. The blue shift of EL spectra upon excitation increase reported previously is explained by the increase of penetration depth of the holes and specific depth distribution of the defects of different kinds.

AUTOCOMMUTATORS AND AUTO-BELL GROUPS

Let x be an element of a group G and be an automorphism of G. Then for a positive integer n, the autocommutator <TEX>$[x,_n{\alpha}]$</TEX> is defined inductively by <TEX>$[x,{\alpha}]=x^{-1}x^{\alpha}=x^{-1}{\alpha}(x)$</TEX> and <TEX>$[x,_{n+1}{\alpha}]=[[x,_n{\alpha}],{\alpha}]$</TEX>. We call the group G to be n-auto-Engel if <TEX>$[x,_n{\alpha}]=[{\alpha},_nx]=1$</TEX> for all <TEX>$x{\in}G$</TEX> and every <TEX>${\alpha}{\in}Aut(G)$</TEX>, where <TEX>$[{\alpha},x]=[x,{\alpha}]^{-1}$</TEX>. Also, for any integer <TEX>$n{\neq}0$</TEX>, 1, a group G is called an n-auto-Bell group when <TEX>$[x^n,{\alpha}]=[x,{\alpha}^n]$</TEX> for every <TEX>$x{\in}G$</TEX> and each <TEX>${\alpha}{\in}Aut(G)$</TEX>. In this paper, we investigate the properties of such groups and show that if G is an n-auto-Bell group, then the factor group <TEX>$G/L_3(G)$</TEX> has finite exponent dividing 2n(n-1), where <TEX>$L_3(G)$</TEX> is the third term of the upper autocentral series of G. Also, we give some examples and results about n-auto-Bell abelian groups.

Weakly Linearly Compact Topological Abelian Groups

In this paper, we study linearly topological groups. We introduce the notion of a weakly linearly compact group, which generalizes the notion of a weakly separable group, and examine the main properties of such groups. For weakly linearly compact groups, we construct the character theory and present an algebraic characterization of some classes of such groups. Some well-known theorems for periodic Abelian groups are generalized for the case of linearly discrete, topological Abelian groups; for linearly compact and linearly discrete topological Abelian groups, we also construct the character theory and study some important properties of linearly discrete groups. For linearly discrete, topological Abelian groups, we analyze the splittability condition (Theorem 3.12) and present the characteristic condition of decomposability of a discrete group G into the direct sum of rank-1 groups. We also present an algebraic characterization of linearly compact groups. We introduce the notion of a weakly linearly compact, topological Abelian group, which generalizes the notion of a weakly separable Abelian group, and examine some properties of such groups. These groups are a generalization of fibrous Abelian groups introduced by Vilenkin. We give an algebraic characterization of divisible, weakly locally compact Abelian groups that do not contain nonzero elements of finite order (Proposition 7.9). For weakly locally compact Abelian groups, we construct universal groups.

On Abelian Groups Close to E-Solvable Groups

E-nilpotent and E-solvable Abelian groups are studied. The properties of such groups are studied, and examples illustrating the differences and connections between the investigated classes of groups are presented. The structure of E-solvable periodical, completely decomposable, coperiodic, split mixed, and other groups is shown.

ACTIONS, LENGTH FUNCTIONS, AND NON-ARCHIMEDEAN WORDS

In this paper we survey recent developments in the theory of groups acting on Λ-trees. We are trying to unify all significant methods and techniques, both classical and recently developed, in an attempt to present various faces of the theory and to show how these methods can be used to solve major problems about finitely presented Λ-free groups. Besides surveying results known up to date we draw many new corollaries concerning structural and algorithmic properties of such groups.

SEMI-PRIMITIVE ROOT MODULO n

Consider a multiplicative group of integers modulo n, denoted by Z n. Any element a 2 Z n is said to be a semi-primitive root if the order of a modulo n is (n)=2, where (n) is the Euler phi-function. In this paper, we classify the multiplicative groups of integers having semi-primitive roots and give interesting properties of such groups. Given a positive integer n, the integers between 1 and n which are coprime ton form a group with multiplication modulon as the operation (4); it is denoted by Z n and is called the multiplicative group of integers modulo n. For any integer a coprime to n, Euler's theorem states that a (n) 1 mod n, where (n) is the Euler phi-function (1), that is, the number of elements in Z n and a is said to be a primitive root modulo n if the order of a modulo n is equal to (n). It is well known (5) that Z n has a primitive root, equivalently, Z n is cyclic if and only if n is equal to 1, 2, 4, p k , or 2p k where p k is a power of an odd prime number. This leaves us questions about Z n that does not possess any primitive roots.

Unusual compression behavior ofTiO2polymorphs from first principles

The physical mechanisms behind the reduction in the bulk modulus of a high-pressure cubic ${\text{TiO}}_{2}$ phase are revealed by first-principles calculations. An unusual and abrupt change occurs in the dependence of energy on pressure at 43 GPa, indicating a pressure-induced phase transition from columbite ${\text{TiO}}_{2}$ to a modified fluorite ${\text{TiO}}_{2}$ with a $Pca21$ symmetry. Oxygen atom displacement in $Pca21$ ${\text{TiO}}_{2}$ unexpectedly reduces the bulk modulus by 34% relative to fluorite ${\text{TiO}}_{2}$. This discovering provides a direct evidence for understanding the compressive properties of such groups of homologous materials.

On the socles of fully invariant subgroups of Abelian p-groups

The classification of the fully invariant subgroups of a reduced Abelian p-group is a difficult long-standing problem when one moves outside of the class of fully transitive groups. In this work we restrict attention to the socles of fully invariant subgroups and introduce a new class of groups which we term socle-regular groups; this class is shown to be large and strictly contains the class of fully transitive groups. The basic properties of such groups are investigated but it is shown that the classification of even this simplified class of groups, seems extremely difficult.

Equational Noetherianness of rigid soluble groups

A group G is said to be rigid if it contains a normal series of the form G = G 1 > G 2 > … > G m > G m + 1 = 1, whose quotients G i /G i + 1 are Abelian and are torsion free as right Z[G/G i ]-modules. In studying properties of such groups, it was shown, in particular, that the above series is defined by the group uniquely. It is known that finitely generated rigid groups are equationally Noetherian: i.e., for any n, every system of equations in x 1, …, x n over a given group is equivalent to some of its finite subsystems. This fact is equivalent to the Zariski topology being Noetherian on G n , which allowed the dimension theory in algebraic geometry over finitely generated rigid groups to have been constructed. It is proved that every rigid group is equationally Noetherian.

Divisible rigid groups

A soluble group G is rigid if it contains a normal series of the form G = G1 > G2 > � > Gp > Gp+1 = 1, whose quotients Gi/Gi+1 are Abelian and are torsion-free as right Z[G/Gi]-modules. The concept of a rigid group appeared in studying algebraic geometry over groups that are close to free soluble. In the class of all rigid groups, we distinguish divisible groups the elements of whose quotients Gi/Gi+1 are divisible by any elements of respective groups rings Z[G/Gi]. It is reasonable to suppose that algebraic geometry over divisible rigid groups is rather well structured. Abstract properties of such groups are investigated. It is proved that in every divisible rigid group H that contains G as a subgroup, there is a minimal divisible subgroup including G, which we call a divisible closure of G in H. Among divisible closures of G are divisible completions of G that are distinguished by some natural condition. It is shown that a divisible completion is defined uniquely up to G-isomorphism.

Automatically presented groups

We introduce the notions of automatically presented groups and piecewise automatically presented groups. We show that if G is a piecewise automatically presented group satisfying the property T of Kazhdan, then G is finite. We prove that if G is amenable and finitely presented, then G is virtually abelian. We give further restrictions for a group to be piecewise automatically presented and study properties of such groups. We also give examples of automatically presented groups.

A Topological Study of Some Groups Arising from Cellular Quotients

Kim and Kostrikin constructed in [8, 9] a tessellation on the boundary of a polyhedral 3-cell consisting of 8n pentagons (n ≥ 1). The tessellation produces a family of closed connected orientable 3-manifolds (denoted by M1(n) in the quoted papers) with spines corresponding to certain presentations of their fundamental groups. We investigate the topological and algebraic properties of such groups together with their derived quotients and split extensions, and completely classify the considered manifolds.