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Difference between revisions of "Verbal product"

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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. [[Free product of groups|Free product of groups]]), $V$ is some set of words, $V(F)$ is the verbal $V$-subgroup (cf. [[Verbal subgroup|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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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><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) pp. 412</TD></TR></table>
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<table>
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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]]

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=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