Convex Subgroups of Partially Right-Ordered Groups

We study into the question of whether a partial order can be induced from a partially right-ordered group onto a space of right cosets of w.r.t. some subgroup of . Examples are constructed showing that the condition of being convex for in is insufficient for this. A necessary and sufficient condition (in terms of a subgroup and a positive cone of ) is specified under which an order of can be induced onto . Sufficient conditions are also given. We establish properties of the class of partially right-ordered groups for which is partially ordered for every convex subgroup , and properties of the class of groups such that is partially ordered for every partial right order on and every subgroup that is convex under .     

Frobenius Groups Generated by Quadratic Elements

An automorphism of a group X is said to be quadratic if there exist integers and such that for any . If is a Frobenius group then an element is said to be quadratic if induces, by conjugation in the core of , a quadratic automorphism. By definition, a group H acts on a group F freely if for and only with or . It is proved that a Frobenius group generated by two quadratic elements is finite and its core is commutative. In particular, any Frobenius group generated by two elements of order at most 4 is finite. Also we argue that a Frobenius group with finitely generated soluble core is finite. The results mentioned are used to show that a group acting freely on an Abelian group is finite if it is generated by elements of order 3, and the order of a product of every two elements of order 3 in is finite.     

Superlocals in Symmetric and Alternating Groups

On Aschbacher's definition, a subgroup N of a finite group is called a -superlocal for a prime if . We describe the -superlocals in symmetric and alternating groups, thereby resolving part way Problem 11.3 in the Kourovka Notebook [3].     

Levi Classes Generated by Nilpotent Groups

Let be a class of all groups G for which the normal closure (x) ^G of every element x belongs to a class . is a Levi class generated by . Let and ₀ be classes of finitely generated nilpotent groups and of torsion-free, finitely generated, nilpotent groups, respectively. We prove that and , and so and . It is shown that quasivarieties and are closed under free products, and that each contains at most one maximal proper subquasivariety. It is also proved that is closed under free products if so is .     

Symmetry of Sections in Fields of Formal Power Series and a Non-Standard Real Line

Let be a field of formal power series with real coefficients, whose supports are well ordered subsets of an Abelian group of cardinality strictly less than . For , we give criteria of a section being symmetric and of a symmetric section being Dedekind. It is proved that an -saturated non-standard real line is isomorphic to some field of the form . For , some consequences are inferred regarding symmetric sections, and the cofinality of banks of the sections.     

Automorphisms of Aschbacher Graphs

If a regular graph of valence and diameter has vertices, then , which was proved by Moore (cf. [1]). Graphs for which this non-strict inequality turns into an equality are called Moore graphs. Such have an odd girth equal to . The simplest example of a Moore graph is furnished by a -triangle. Damerell proved that a Moore graph of valence has diameter 2. In this case , the graph is strongly regular with and , and the valence is equal to 3 (Peterson's graph), to 7 (Hoffman–Singleton's graph), or to 57. The first two graphs are of rank 3. Whether a Moore graph of valence exists is not known; yet, Aschbacher proved that the Moore graph with will not be a rank 3 graph. We call the Moore graph with the Aschbacher graph. Cameron showed that such cannot be vertex transitive. Here, we treat subgraphs of fixed points of Moore graph automorphisms and an automorphism group of the hypothetical Aschbacher graph for the case where that group contains an involution.     

Equational Theories for Classes of Finite Semigroups

It is proved that there exists an infinite sequence of finitely based semigroup varieties such that, for all i, an equational theory for and for the class of all finite semigroups in is undecidable while an equational theory for and for the class of all finite semigroups in is decidable. An infinite sequence of finitely based semigroup varieties is constructed so that, for all i, an equational theory for and for the class of all finite semigroups in is decidable whicle an equational theory for and for the class of all finite semigroups in is not.     

Almost periodic compactifications of group extensions

Let and be groups and let be an extension of by . Given a property of group compactifications, one can ask whether there exist compactifications and of N and K such that the universal -compactification of G is canonically isomorphic to an extension of by . We prove a theorem which gives necessary and sufficient conditions for this to occur for general properties and then apply this result to the almost periodic and weakly almost periodic compactifications of G.     

Relatively Hyperimmune Relations on Structures

Let be a computable structure and let R be an additional relation on its domain. We establish a necessary and sufficient condition for the existence of an isomorphic copy of such that the image of R is h-simple (h-immune) relative to .     

Factorizations of One-Generated Composition Formations

A non-empty formation of finite groups is said to be solubly saturated, or we call it a composition formation, if every finite group G having a normal subgroup N such that belongs to . An intersection of all composition formations containing a given group G is denoted cformG. Conditions are described under which has the form , where .     

The Lattice of Quasivarieties of Torsion-Free Metabelian Groups

Assume that a quasivariety of groups contains a non-Abelian free metabelian group and a non-Abelian free nilpotent group of class 2. It is proved that the lattice of quasivarieties contained in is infinite and non-modular.     

A Question in the Theory of Totally Local Formations of Finite Groups

It is proved that all proper totally local subformations of a non one-generated totally local formation of finite groups are one-generated iff coincides with a formation of all soluble -groups, where ||=2.     

Some estimates for the oscillation of the deformation gradient

As a measure of deformation we can take the difference D - R, where D is the deformation gradient of the mapping and R is the deformation gradient of the mapping , which represents some proper rigid motion. In this article, the norm is estimated by means of the scalar measure e( ) of nonlinear strain. First, the estimates are given for a deformation W ^1,p() satisfying the condition . Then we deduce the estimate in the case that (x) is a bi-Lipschitzian deformation and .     

Test Rank for Some Free Polynilpotent Groups

We prove a theorem on possible test rank values for groups of the form . It is shown that test rank of a free polynilpotent group is equal to or , for any and every collection of classes. Moreover, for and .     

Lattices of interpretability types of varieties

Let be the set of all primes, the field of all algebraic numbers, and Z the set of square-free natural numbers. We consider partially ordered sets of interpretability types such as , and , where AD is a variety of -divisible Abelian groups with unique taking of the pth root _p(x) for every p , is a variety of -modules over a normal field , contained in , and G_n is a variety of n-groupoids defined by a cyclic permutation (12 ...n). We prove that , and are distributive lattices, with and where ub and ub_f are lattices (w.r.t. inclusion) of all subsets of the set and of finite subsets of , respectively.Deceased.__________Translated from Algebra i Logika, Vol. 44, No. 2, pp. 198–210, March–April, 2005.     

An Analog for the Frattini Factorization of Finite Groups

Using the classification of finite simple groups, we prove that if H is an insoluble normal subgroup of a finite group G, then H contains a maximal soluble subgroup S such that G=HN_G(S). Thereby Problem 14.62 in the Kourovka Notebook is given a positive solution. As a consequence, it is proved that in every finite group, there exists a subgroup that is simultaneously a -projector and a -injector in the class, , of all soluble groups.