Geodesic
On free products of groups
Sbornik. Mathematics, Tome 8 (1969) no. 4, pp. 593-597 
A. L. Shmel'kin. On free products of groups. Sbornik. Mathematics, Tome 8 (1969) no. 4, pp. 593-597. http://geodesic.mathdoc.fr/item/SM_1969_8_4_a3/
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     title = {On free products of groups},
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Voir la notice de l'article provenant de la source Math-Net.Ru

Let $F=\prod^*G_i$ be a free product with a normal subgroup $R$, and let $V(R)$ be a verbal subgroup of $R$. The main result of this paper asserts that when $R$ is contained in the Cartesian subgroup of $F$, $F/V(R)$ is embeddable in the verbal $V$-wreath product of a $\mathfrak B$-free group by $F/R$ (here $\mathfrak B$ is the variety defined by the laws $V$). This embedding reduces, to a great extent, the study $F/V(R)$ to that of $F/R$ and $R/V(R)$. New as well as known results about $F/V(R)$ are obtained as corollaries of the above-mentioned theorem. Bibliography: 7 titles.

[1] W. Magnus, “On a theorem of Marshall Hall”, Ann. Math., 40 (1939), 764–769 | DOI | MR

[2] A. L. Shmelkin, “Spleteniya i mnogoobraziya grupp”, Izv. AN SSSR, seriya matem., 29 (1965), 149–170

[3] A. L. Shmelkin, “Zamechaniya k moei rabote “Spleteniya i mnogoobraziya grupp””, Izv. AN SSSR, seriya matem., 31 (1967), 443–444

[4] O. N. Golovin, “Nilpotentnye proizvedeniya grupp”, Matem. sb., 27(69) (1950), 427–454 | MR | Zbl

[5] M. Krasner, L. Kalonjnine, “Produit complet des groupes de permutation et le problème d'extension de groups, III”, Acta Sci. Math. Szeged, 14 (1951), 69–82 | MR | Zbl

[6] H. Neumann, Varieties of groups, Springer, Berlin, Heidelberg, New York, 1967 | MR

[7] K. W. Gruenberg, “Residual properties of infinite soluble groups”, Proc. London Math. Soc., 7:25 (1957), 29–62 | DOI | MR | Zbl