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A translation in an algebraic signature is a finite conjunction of equations in one variable. On a quasivariety K, a translation τ naturally induces a deductive system, called the τ-assertional logic of K. Two quasivarieties are τ-assertionally equivalent if they have the same τ-assertional logic. This paper is a study of assertional equivalence. It characterizes the quasivarieties equivalent to ones with various desirable properties, such as τ-regularity (a general form of point regularity). Special attention is paid to structural properties of quasivarieties that are assertionally equivalent to their varietal closures under an indicated translation. [ABSTRACT FROM AUTHOR]

The commutator length "cl(G)" of a group G is the least natural number c such that every element of the derived subgroup of G is a product of c commutators. We give an upper bound for cl(G) when G is a d-generator nilpotent-by-abelian-by-finite group. Then, we give an upper bound for the commutator length of a soluble-by-finite linear group over C that depends only on d and the degree of linearity. For such a group G, we prove that cl(G) is less than k(k + 1)/2 + 12d3 + o(d2), where k is the minimum number of generators of (upper) triangular subgroup of G and o(d2) is a quadratic polynomial in d. Finally we show that if G is a soluble-by-finite group of Prüffer rank r then cl(G) ≤ r(r + 1)/2 + 12r3 + o(r2), where o(r2) is a quadratic polynomial in r. [ABSTRACT FROM AUTHOR]

We study the representations of non-commutative universal lattices and use them to compute lower bounds of the τ-constant for the commutative universal lattices G d, k =SL d (ℤ[ x 1,..., x k ]), for d≥3 with respect to several generating sets. As an application we show that the Cayley graphs of the finite groups $\text{SL}_{3k}(\mathbb{F}_{p})$ can be made expanders with a suitable choice of generators. This provides the first example of expander families of groups of Lie type, where the rank is not bounded and provides counter examples to two conjectures of A. Lubotzky and B. Weiss. [ABSTRACT FROM AUTHOR]

In this paper, some results related to the subnormal join property (SJP) for periodic CC-groups are given. Also, we examine Θ (H, 픛), a subgroup of G, which is a minimal non-CC-p-group, where 픛 is the class of CC-groups. We prove that G satisfies SJP. [ABSTRACT FROM AUTHOR]

Given a finite group G we show that Dress and Siebeneicher's ring of G-typical Witt vectors on the Lazard ring, that is, on the polynomial ring on countably many indeterminates over the integers, embeds as a subring of the unitary cobordism ring of G-manifolds. We also show that the ring of G-typical Witt vectors on the Lazard ring embeds as a subring of the ring of homotopy groups of the G-fixed point spectrum of the spectrum MU representing cobordism. The above results are derived by exploiting the interaction between restriction, additive transfer and multiplicative transfer. This interaction is described by two Mackey functors satisfying a distributivity relation encoded in a formalism developed by Tambara. [ABSTRACT FROM AUTHOR]

Let Γ be a finitely generated infinite group. Denote by K(Γ) the FC-centre of Γ, i.e. the subgroup of all elements of Γ having only finitely many conjugates in Γ. Let QI(Γ) denote the group of quasi-isometries of Γ with respect to a word metric. We prove that the natural homomorphism θγ: Aut (Γ) → QI (Γ) is a monomorphism only if K(Γ) equals the centre Z(Γ) of Γ. The converse holds K(Γ) = Z(Γ) is torsion-free. When K (Γ) is finite we show that θ. is a monomorphism where Γ̄ = Γ K(Γ). We apply this criterion to a number of classes of groups arising in combinatorial and geometric group theory. [ABSTRACT FROM AUTHOR]

We obtain a test for the unsimplicity of an infinite group. [ABSTRACT FROM AUTHOR]

A finite commutative chain ring is a finite commutative ring whose ideals form a chain. Let R be a finite commutative ring with maximal ideal M and characteristic pn such that R/M≅GF(pr) and pR=Me, e⩽s, where s is the nilpotency of M. When (p−1)∤e, the structure of the group of units R× of R has been determined; it only depends on the parameters p,n,r,e,s. In this paper, we give an algorithmic method which allows us to compute the structure of R× when (p−1) | e; such a structure not only depends on the parameters p,n,r,e,s, but also on the Eisenstein polynomial which defines R as an extension over the Galois ring GR(pn,r). In the case (p−1)∤e, we strengthen the known result by listing a set of linearly independent generators for R×. In the case (p−1) | e but p∤e, we determine the structure of R× explicitly. [Copyright &y& Elsevier]

Dedicated to the memory of Professor A. I. KostrikinThe main problem under discussion is to determine, for quasi-simple groups of Lie type G, irreducible representations φ of G that remain irreducible under reduction modulo the natural prime p. The method is new. It works only for p >3 and for representations φ that can be realized over an unramified extension of Qp, the field of p -adic numbers. Under these assumptions, the main result says that the trivial and the Steinberg representations of G are the only representations in question provided G is not of type A1. This is not true for G=SL(2, p). The paper contains a result of independent interest on infinitesimally irrreducible representations ρ of G over an algebraically closed field of characteristic p. Assuming that G is not of rank 1 and G≠ G2(5), it is proved that either the Jordan normal form of a root element contains a block of size d with 1

We prove that the fundamental group of the double of the figure-eight knot exterior admits a faithful discrete representation into SO(4, 1; R), for which the image group is separable on its geometrically finite subgroups. [ABSTRACT FROM PUBLISHER]

Let $A$ be a connected finite dimensional $k$-algebra, and let $M$ be a nonzero decomposable $A$-module such that the one-point extension $A[M]$ is quasitilted. We show here that every nonzero indecomposable direct summand of $M$ is directing and $A$ is a tilted algebra. [ABSTRACT FROM AUTHOR]

The paper considers finite rank torsion-free rings, i.e, subrings of finite-dimensional rational algebras. The main result of the paper is the characterization of finite rank torsion-free rings with finite automorphism groups. Nontrivial examples of rings R with AutR finite have been constructed. [ABSTRACT FROM AUTHOR]

Discusses the study of infinite dimensional linear groups. Generalizations of irreducible groups; Proofs and theorems.

Let a group G contain a Carter subgroup of odd order. It is shown that every composition factor of G either is Abelian or is isomorphic to L(3), n ≥ 1 . Moreover, if 3 does not divide the order of a Carter subgroup, then G solvable. [ABSTRACT FROM AUTHOR]

B. Hartley in [4] introduced the class of barely transitive groups. By definition, a group of permutations G on an infinite set X is called barely transitive if G itself is transitive on X while every orbit of every proper subgroup of G is finite. If G is locally finite and G2`G then the theorem of B. Love, proved in [4], shows that G is a locally nilpotent p-group of Heineken-Mohamed type. However, it is not known if perfect barely transitive locally nilpotent p-groups exist. Obviously this is a more general question than the corresponding one about perfect minimal non FC-groups (see below for definition). In this work it will be shown that a barely transitive locally nilpotent p-group cannot be perfect if the stabilizer of a point is hypercentral and solvable. [ABSTRACT FROM AUTHOR]

The norm of a group G is the subgroup of elements of G which normalise every subgroup of G. We shall denote it κ(G). An ascending series of subgroups κi(G) in G may be defined recursively by: κ0(G) = 1 and, for i ≥ 0, κi+1(G)/κi(G) = κ(G/κi(G)). For each i, the section κi+1(G)/κi(G) clearly contains the centre of the group G/κi(G). A result of Schenkman [8] gives a very close connection between this norm series and the upper central series: ζi(G) ⊆ κi(G) ⊆ ζ2i(G). 1991 Mathematics Subject Classification 20E15. [ABSTRACT FROM AUTHOR]

Let G be a finite group acting faithfully and transitively on a set Ω of size m, and let E = {x1, …, xk} be a generating set for G with x1x2…xk = 1. If x ∈ G has cycles of length r1, …, rl in its action on Ω, define ind(x)=∑1l(ri−1). Then the genus g = g(G, Ω, E) is defined by 2(m+g−1)=∑1kind(xi), 1991 Mathematics Subject Classification 20B25, 20G40, 30F99. [ABSTRACT FROM AUTHOR]

Interactions between IgG Fc and its receptors (FcγRs) have been shown to augment broadly neutralizing Ab-mediated (BnAb-mediated) protection from simian-human immunodeficiency virus (SHIV) challenge. In the current issue of the JCI, Parsons and collaborators compared the BnAb PGT121 with a version engineered to have impaired FcγR binding for their ability to protect macaques from an intravenous challenge with SHIV-infected cells as well as to treat already infected animals. Unexpectedly, and in contrast to previous studies, both versions of the Ab were equally able to prevent infection and decrease viral loads in infected animals. Thus, FcγR engagement does not always improve the in vivo antiviral activity of BnAbs. [ABSTRACT FROM AUTHOR]

A group acting on an abelian group is finite provided that it is generated by a class of conjugate elements such that every two elements of the class generate a finite subgroup that acts freely. [ABSTRACT FROM AUTHOR]

Let be a finite group, whose order has prime divisors. In this paper, we prove that if has a -complement for prime divisors of and has no section isomorphic to . Then is solvable, which generalizes a theorem of Hall. [ABSTRACT FROM AUTHOR]

LetGbe an arbitraryFC-group, letRbe its locally soluble radical, and letL/R=L(G/R). We prove that, forN?G,G/Nis residually finite ifRNL. [ABSTRACT FROM AUTHOR]