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Verbal product

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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.


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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) pp. 412
How to Cite This Entry:
Verbal product. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Verbal_product&oldid=32423
This article was adapted from an original article by O.N. Golovin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article