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If f:G → H is a group homomorphism and p,q are the projections from the free product G*H onto its factors G and H respectively, let the group $_{f} \subseteq G * H$ be the equalizer of fp and q:G*H → H. Then p restricts to an epimorphism $p_{f} {= p |}_{f} :_{f} \to G$ . A right inverse (section) $G \to_{f}$ of $p_{f}$ is called a coaction on G. In this paper we study $_{f}$ and the sections of $p_{f}$ . We consider the following topics: the structure of $_{f}$ as a free product, the restrictions on G resulting from the existence of a coaction, maps of coactions and the resulting...

Equations in groups, with special emphasis on localization and torsion - II

Ribenboim, P. (1987)

Portugaliae mathematica

Equivalence élémentaire entre groupes finis-par-abéliens de type fini.

Francis Oger (1982)

Commentarii mathematici Helvetici

Equivalence of the strengthened Hanna Neumann conjecture and the amalgamated graph conjecture.

Warren Dicks (1994)

Inventiones mathematicae

Equivalence relations induced by some locally compact groups of homeomorphisms of $2^{ℕ}$

B. Majcher-Iwanow (2005)

Colloquium Mathematicae

Let T be a locally finite rooted tree and B(T) be the boundary space of T. We study locally compact subgroups of the group TH(B(T)) = ⟨Iso(T),V⟩ generated by the group Iso(T) of all isometries of B(T) and the group V of Richard Thompson. We describe orbit equivalence relations arising from actions of these groups on B(T).

Equivalent Expressions of Direct Sum Decomposition of Groups1

Kazuhisa Nakasho, Hiroyuki Okazaki, Hiroshi Yamazaki, Yasunari Shidama (2015)

Formalized Mathematics

In this article, the equivalent expressions of the direct sum decomposition of groups are mainly discussed. In the first section, we formalize the fact that the internal direct sum decomposition can be defined as normal subgroups and some of their properties. In the second section, we formalize an equivalent form of internal direct sum of commutative groups. In the last section, we formalize that the external direct sum leads an internal direct sum. We referred to [19], [18] [8] and [14] in the...