Namespaces
Variants
Actions

Difference between revisions of "Verbal product"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(TeX)
Line 1: Line 1:
''of groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096570/v0965701.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096570/v0965702.png" />''
+
{{TEX|done}}
 +
''of groups $G_i$, $i\in I$''
  
The quotient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096570/v0965703.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096570/v0965704.png" /> is the free product of the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096570/v0965705.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096570/v0965706.png" /> (cf. [[Free product of groups|Free product of groups]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096570/v0965707.png" /> is some set of words, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096570/v0965708.png" /> is the verbal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096570/v0965709.png" />-subgroup (cf. [[Verbal subgroup|Verbal subgroup]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096570/v09657010.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096570/v09657011.png" /> is the Cartesian subgroup (i.e. the kernel of the natural epimorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096570/v09657012.png" /> 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.
+
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.
  
  

Revision as of 19:59, 12 July 2014

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) 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