# Verbal product

of groups $G_i$, $i\in I$

The quotient group $F/(V(F)\cap C)$, where $F$ is the free product of the groups $G_i$, $i\in I$ (cf. Free product of groups), $V$ is some set of words, $V(F)$ is the verbal $V$-subgroup (cf. Verbal subgroup) of $F$, and $C$ is the Cartesian subgroup (i.e. the kernel of the natural epimorphism of $F$ onto the direct product of these groups). As an operation on a class of groups, the verbal product is associative and, within the corresponding variety of groups, it is also free.