The free product of \(\Gamma\)-torsionfree groups. (Russian) Zbl 0631.06009

Theorem 1: The free product of two right-orderable \(\Gamma\)-torsionfree groups is right-orderable \(\Gamma\)-torsionfree. (The group G is said to be \(\Gamma\)-torsionfree if \(xx^{g_ 1}...x^{g_ n}=e\) for any elements \(x,g_ 1,...,g_ n\) in G implies \(x=e.)\) This gives a partial answer to a question of R. Mura and A. Rhemtulla [Orderable Groups (Lect. Notes Pure Appl. Math. 27) (1977; Zbl 0452.06011)] about free products of \(\Gamma\)-torsionfree groups. A new proof (Theorem 2) is also given of an old theorem of A. A. Vinogradov [Mat. Sb., Nov. Ser. 25(67), 163-168 (1949; Zbl 0038.159)] to the effect that the free product of two orderable groups is orderable.
Reviewer: J.D.Macdonald


06F15 Ordered groups
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
20F60 Ordered groups (group-theoretic aspects)
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