Narrow Results

Let $FG$ be the group algebra of a residually torsion free nilpotent group $G$ over a field $F$ of characteristic $0$. If $(x,y)$ is any pair of noncommuting elements of $G$, we show that for any rational number $r$ with $r\ne 0,\pm 1$, the subgroup $〈1+rx,1+ry,1+rxy〉$ is free of rank $3$ in the Malcev–Neumann field of fractions of $FG$. This result comes from a method of producing free groups of rank three in rational quaternions. In general, if a division ring $D$ has dimension $d^2$ over its center $Z$, and if the transcendence degree of $Z$ over its prime field ${𝔽}_p$ is $\ge d^2+3$, we present a method to construct free subgroups of rank three.

Large order cryptographically suitable quasigroups have important applications in the development of crypto-primitives and cryptographic schemes. These present new perspectives of cryptography and information security. From algebraic point of view polynomial completeness is one of the most important characteristic for cryptographically suitable quasigroups. In this paper, we propose four different methods to construct polynomially complete quasigroups of any order $n\ge 5$. First method is based on a starting quasigroup of same order, second method is based on a particular permutation of $S{S}_n$ and third and fourth methods are based on products of lower order quasigroups. In the last case, all quasigroups and their isotopes are polynomially complete. We also develop and implement an algorithm to derive a permutation for a given permutation of $S{S}_n,n\ge 5,$ so that they generate whole $S{S}_n$.

Here we are concerned on Bogomolov's problem for hypersurfaces; we give a geometric lower bound for the height of a hypersurface of (i.e. without condition on the field of definition of the hypersurface) which is not a translate of an algebraic subgroup of . This is an analogue of a result of F. Amoroso and S. David who give a lower bound for the height of non-torsion hypersurfaces defined and irreducible over the rationals.

Let ${\lambda }_1(n)$ denote the least invariant factor in the invariant factor decomposition of the multiplicative group $M_n={(\mathbb{Z}/n\mathbb{Z})}^{\times }$. We give an asymptotic formula, with order of magnitude $x/\sqrt{logx}$, for the counting function of those integers $n$ for which ${\lambda }_1(n)\ne 2$. We also give an asymptotic formula, for any even $q\ge 4$, for the counting function of those integers $n$ for which ${\lambda }_1(n)=q$. These results require a version of the Selberg–Delange method whose dependence on certain parameters is made explicit, which we provide in Appendix A. As an application, we give an asymptotic formula for the counting function of those integers $n$ all of whose prime factors lie in an arbitrary fixed set of reduced residue classes, with implicit constants uniform over all moduli and sets of residue classes.

We examine two counting problems that seem very group-theoretic on the surface but, on closer examination, turn out to concern integers with restrictions on their prime factors. First, given an odd prime $q$ and a finite abelian $q$-group $H$, we consider the set of integers $n\le x$ such that the Sylow $q$-subgroup of the multiplicative group ${(\mathbb{Z}/n\mathbb{Z})}^{\times }$ is isomorphic to $H$. We show that the counting function of this set of integers is asymptotic to $Kx{(loglogx)}^{\ell }/{(logx)}^{1/(q-1)}$ for explicit constants $K$ and $\ell$ depending on $q$ and $H$. Second, we consider the set of integers $n\le x$ such that the multiplicative group ${(\mathbb{Z}/n\mathbb{Z})}^{\times }$ is “maximally non-cyclic”, that is, such that all of its prime-power subgroups are elementary groups. We show that the counting function of this set of integers is asymptotic to $Ax/{(logx)}^{1-\xi }$ for an explicit constant $A$, where $\xi$ is Artin’s constant. As it turns out, both of these group-theoretic problems can be reduced to problems of counting integers with restrictions on their prime factors, allowing them to be addressed by classical techniques of analytic number theory.

We give a short proof of Manin–Mumford in the multiplicative group based on the pigeon-hole principle and a simple structure theorem for tori contained in subvarieties of ${𝔾}_m^n$ due to Bombieri and Gubler. The arguments appear to be new and are applicable in other situations. Our proof is quite elementary and we compare it to previous proofs that rely on machinery such as intersection theory or $o$-minimality.

Let $S(n)$ denote the least primary factor in the primary decomposition of the multiplicative group $M_n={(\mathbb{Z}/n\mathbb{Z})}^{\times }$. We give an asymptotic formula, with order of magnitude $x/{(logx)}^{1/2}$, for the counting function of those integers n for which $S(n)\ne 2$. We also give an asymptotic formula, for any prime power q, for the counting function of those integers n for which $S(n)=q$. This group-theoretic problem can be reduced to problems of counting integers with restrictions on their prime factors, allowing it to be addressed by classical techniques of analytic number theory.

On the projectively extended euclidean plane of a fixed triangle, point-wise binary operations and transformations described in barycentric coordinates arise in a geometric way. We apply these operations over the points of a standard Bézier parabola and a weigh point, and get families of associated curves. There exist two geometric multiplicative group structures. We describe the trajectories of a rational Bézier curve by means of group orbits of its points. We calculate the curvature of a smooth curve in barycentrics and apply the formula to a rational Bézier curve.