We classify all subgroups of SO(3) that are generated by two elements, each a rotation of finite order, about axes separated by an angle that is a rational multiple of π. In all cases we give a presentation of the subgroup. In most cases the subgroup is the free product, or the amalgamated free product, of cyclic groups or dihedral groups. The relations between the generators are all simple con...

For any field K, we determine the quasi-simple subgroups G of the Chevalley group F4ðKÞ which are generated by a class S of so-called abstract transvection subgroups of G such that any member of S is contained in a long root subgroup of F4ðKÞ. First, we construct a polar space with point set S, which is embedded in the F4-geometry. This yields that a conjugate of G is contained in a classical s...

We address two properties for Abelian topological groups: “every closed subgroup is dually closed” and “every closed subgroup is dually embedded.” We exhibit a pair of topological groups such that each has both of the properties and the product has neither, which refutes a remark of N. Noble. These examples are the additive group of integers topologized with respect to a convergent sequence as ...

Let Hg be a genus g handlebody, and Tg = ∂Hg a closed connected orientable surface. In this paper we find a finite set of generators for Eg 2 , the subgroup of PMCG2(Tg) consisting of the isotopy classes of homeomorphisms of Tg which admit an extension to the handlebody keeping a properly embedded trivial arc fixed. This subgroup turns out to be important for the study of knots in closed 3-mani...

Let X be a topological abelian group and H subgroup of X. We find conditions under which local quasi-convexity both X/H results in the same property for This is true instance if precompact, or metrizable dually embedded also either discrete bounded torsion. give some general principles point out errors we have found existing literature on this problem.

Let ı : T p → Rp+2, p ≥ 1, be an embedded p-torus into the Euclidean space Rp+2, and Eı be the subgroup of MT p , the mapping class group of T , whose elements extend over Rp+2 as self-diffeomorphisms. We show that for any unknotted embedding ı : T p → Rp+2, Eı is a subgroup of MT p of index 2−1. We also provide a matrix characterization. This applies to showing that for any expanding map φ : T...

objective: in this article, a numerical study is conducted on mixing of two fluids in the liquid phase with two different concentrations of a chemical species in the electro-osmotic flow. methods: the base liquid is an electrolyte which flows in a two-dimensional micro-channel having electrically charged walls. lorentz electric force, which is used as stimulating flow factor, is created by appl...