View source for Verbal product

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''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.



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====References====
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Magnus,   A. Karrass,   B. Solitar,   "Combinatorial group theory: presentations in terms of generators and relations" , Wiley (Interscience)  (1966) {{ZBL|0138.25604}}. pp. 412</TD></TR>
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[[Category:Group theory and generalizations]]