The extended modular group $\overline{\Gamma}$ is isomorphic to the amalgamated free product of two dihedral groups $D_2$ and $D_3$ with amalgamation $\mathbb{Z}_2$. This acts on rational numbers transitively. In this study, we obtain elements in that are mappings between given rationals. Also, express these as a word generators. We use interesting relations continued fractions Farey graph.

Let G be a finite group and W be a faithful representation of G over C. The group G acts on the field of rational functions C(W ). The aim of this paper is to give a description of the unramified cohomology group of degree 3 of the field of invariant functions C(W ) in terms of the cohomology of G when G is a group of odd order. This enables us to give an example of a group for which this field...

Nathaniel Dean asks the following: Is it possible to find four nonconcyclic points on the parabola y = x2 such that each of the six distances between pairs of points is rational? We demonstrate that there is a correspondence between all rational points satisfying this condition and orbits under a particular group action of rational points on a fiber product of (three copies of) an elliptic surf...

This study aims to reduce cheating behavior in students by using Rational Emotive Behavior Therapy group counseling. research is a pre-experimental design research, namely One Group Pretest and Posttest Design. The subjects of this were ten third semester the Guidance Counseling program at Ahmad Dahlan University, taken purpsoive sampling technique. instrument used scale. Analysis data effectiv...

A toroidal embedding is defined which does not assume the fan consists of rational cones. For a rational fan, the toroidal embedding is the usual toric variety. If the fan is not rational, the toroidal embedding is in general a quasicompact noetherian locally ringed space which is not a scheme. A divisor theory exists and a class group is defined. A second construction is also carried out which...