Difference between revisions of "Verbal product"
From Encyclopedia of Mathematics
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''of groups $G_i$, $i\in I$'' | ''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. [[ | + | 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==== | ====References==== | ||
<table> | <table> | ||
| − | <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) | + | <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> |
</table> | </table> | ||
[[Category:Group theory and generalizations]] | [[Category:Group theory and generalizations]] | ||
Latest revision as of 19:33, 11 December 2015
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.
Comments
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=33708
Verbal product. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Verbal_product&oldid=33708
This article was adapted from an original article by O.N. Golovin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article