## 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

### MSC:

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

### Citations:

Zbl 0452.06011; Zbl 0038.159
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