# Verbal product

From Encyclopedia of Mathematics

*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

[a1] | W. Magnus, A. Karrass, B. Solitar, "Combinatorial group theory: presentations in terms of generators and relations" , Wiley (Interscience) (1966) Zbl 0138.25604. pp. 412 |

**How to Cite This Entry:**

Verbal product.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Verbal_product&oldid=36889

This article was adapted from an original article by O.N. Golovin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article