Conjugate homeomorphisms of the Rational World.
In this short note, it is shown that if are -connected transversals for a finite subgroup of an infinite group such that the index of in is at least 3 and whenever and then is a normal abelian subgroup of .
Subnormal subgroups possessing connected transversals are briefly discussed.
In this article, conservation rules of the direct sum decomposition of groups are mainly discussed. In the first section, we prepare miscellaneous definitions and theorems for further formalization in Mizar [5]. In the next three sections, we formalized the fact that the property of direct sum decomposition is preserved against the substitutions of the subscript set, flattening of direct sum, and layering of direct sum, respectively. We referred to [14], [13] [6] and [11] in the formalization.
In this paper we study an action of the absolute Galois group on bicolored plane trees. In distinction with the similar action provided by the Grothendieck theory of “Dessins d’enfants” the action is induced by the action of on equivalence classes of conservative polynomials which are the simplest examples of postcritically finite rational functions. We establish some basic properties of the action and compare it with the Grothendieck action.