Generators and relations for wreath products

Generators and defining relations for wreath products of groups are given. Under a certain condition (conormality of generators), they are minimal. Published in Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 7, pp. 997–999, July, 2008.     

Minimal surface as a diagram of sphere rotations

Based on I. N. Vekya's representation of the field of infinitely small (i. s.) bendings of a sphere in terms of analytic functions, we present a new proof of Liebmann's theorem to the effect that the diagram of rotations of i.s. bendings of a sphere is a minimal surface and, conversely, each minimal surface is the diagram of rotations of some i.s. bending of a sphere or of part of it. It is then established that all the minimal surfaces which are non-trivially locally isometric to a given minimal surface constitute an analytic single-parameter family, and explicit expressions for the surfaces of this family are given. The bibliography contains four titles.Translated from Matematicheskie Zametki, Vol. 2, No. 6, pp. 645–656, December, 1967.     

Generators and relations of direct products of semigroups

The purpose of this paper is to give necessary and sufficient conditions for the direct product of two semigroups to be finitely generated, and also for the direct product to be finitely presented. As a consequence we construct a semigroup of order 11 such that is finitely generated but not finitely presented for every finitely generated infinite semigroup . By way of contrast we show that, if and belong to a wide class of semigroups, then is finitely presented if and only if both and are finitely presented, exactly as in the case of groups and monoids.

     

Finite group extensions of irrational rotations

The ergodicity of certain skew products of irrational rotations of the circle with finite groups is established with application to the construction of “well-distributed sequence generators” for finite groups.     

Generators and relations of abelian semigroups and semigroup rings

The object of this paper is the study of the relations of finitely generated abelian semigroups. We give a new proof of the fact that each such semigroup S is finitely presented. Moreover, we show that the number of relations defining S is greater than or equal to the least number of generators of S minus the rank of the associated group of S. If equality holds, we say S is a complete intersection. The main part of this study is devoted to semigroups of natural numbers generated by 3 elements. These semigroups are complete intersections if and only if they are symmetric in the sense of R. Apéry [1]. This result applies to algebraic geometry: An affine space-curve C with the parametric equations x=t^a, y=t^b, z=t^c, a, b, c natural numbers with greatest common divisor 1, is a global idealtheoretic complete intersection, if and only if the semigroup S generated by a, b, c is symmetric.This paper forms part of the author's thesis, submitted at Lousiana State University.The writing of this paper was partially supported by NSF grant GP-6388 in which the author participated as a junior assistant at Purdue University.     

Generators of large subgroups of the unit group of integral group rings

This work is supported in part by NSERC Grant OGP0036631, Canada, and CNPq, Brasil     

The quaternion group as a subgroup of the sphere braid groups

Let n 3. We prove that the quaternion group of order 8 is realisedas a subgroup of the sphere braid group B_n(²) if and only ifn is even. If n is divisible by 4, then the commutator subgroupof B_n(²) contains such a subgroup. Further, for all n 3, B_n(²)contains a subgroup isomorphic to the dicyclic group of order4n.     

Cubature formulas on a sphere that are invariant with respect to a group of dihedron rotations with inversion D 6h

An algorithm of searching for the best (in a sense) cubature formulas on a sphere that are invariant with respect to a group of dihedron rotations with inversion D _6h is developed. This algorithm is applied to find parameters of all the best cubature formulas of this group of symmetry up to the 23rd order of accuracy n. In the course of the study performed, exact values of parameters of the corresponding cubature formulas are found for n ≤ 11, and approximate values are obtained by numerical solving systems of nonlinear algebraic equations by a Newton-type method for other values of n. For the first time, ways of obtaining the best cubature formulas for a sphere are systematically investigated for the case of a group that is not a subgroup of the groups of symmetry of regular polyhedrons.     

Generators and relations for subsemigroups via boundaries in Cayley graphs

Given a finitely generated semigroup S and subsemigroup T of S, we define the notion of the boundary of T in S which, intuitively, describes the position of T inside the left and right Cayley graphs of S. We prove that if S is finitely generated and T has a finite boundary in S then T is finitely generated. We also prove that if S is finitely presented and T has a finite boundary in S then T is finitely presented. Several corollaries and examples are given.