Verbal product

of groups ,

The quotient group , where is the free product of the groups , (cf. Free product of groups), is some set of words, is the verbal -subgroup (cf. Verbal subgroup) of , and is the Cartesian subgroup (i.e. the kernel of the natural epimorphism of 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.

References

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