On free products of groups

A. L. Shmel'kin

Sbornik. Mathematics, Tome 8 (1969) no. 4, pp. 593-597

Résumé

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.