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Conjugacy pinched and cyclically pinched one-relator groups.

Benjamin Fine, Gerhard Rosenberger, Michael Stille (1997)

Revista Matemática de la Universidad Complutense de Madrid

Here we consider two classes of torsion-free one-relator groups which have proved quite amenable to study-the cyclically pinched one-relator groups and the conjugacy pinched one-relator groups. The former is the class of groups which are free products of free groups with cyclic amalgamations while the latter is the class of HNN extensions of free groups with cyclic associated subgroups. Both are generalizations of surface groups. We compare and contrast results in these classes relative to n-freeness,...

Connected transversals -- the Zassenhaus case

Tomáš Kepka, Petr Němec (2000)

Commentationes Mathematicae Universitatis Carolinae

In this short note, it is shown that if A , B are H -connected transversals for a finite subgroup H of an infinite group G such that the index of H in G is at least 3 and H ∩ H u ∩ H v = 1 whenever u , v ∈ G ∖ H and u v - 1 ∈ G ∖ H then A = B is a normal abelian subgroup of G .

Conservation Rules of Direct Sum Decomposition of Groups

Kazuhisa Nakasho, Hiroshi Yamazaki, Hiroyuki Okazaki, Yasunari Shidama (2016)

Formalized Mathematics

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.

Conservative polynomials and yet another action of Gal ( ℚ ¯ / ℚ ) on plane trees

Fedor Pakovich (2008)

Journal de Théorie des Nombres de Bordeaux

In this paper we study an action D of the absolute Galois group Γ = Gal ( ℚ ¯ / ℚ ) on bicolored plane trees. In distinction with the similar action provided by the Grothendieck theory of “Dessins d’enfants” the action D is induced by the action of Γ on equivalence classes of conservative polynomials which are the simplest examples of postcritically finite rational functions. We establish some basic properties of the action D and compare it with the Grothendieck action.

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