Namespaces
Variants
Actions

Difference between revisions of "Central product of groups"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(Tex done)
 
Line 1: Line 1:
A group-theoretical construction. A group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021190/c0211901.png" /> is called a central product of two of its subgroups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021190/c0211902.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021190/c0211903.png" /> if it is generated by them, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021190/c0211904.png" /> for any two elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021190/c0211905.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021190/c0211906.png" /> and if the intersection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021190/c0211907.png" /> lies in its centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021190/c0211908.png" />. In particular, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021190/c0211909.png" /> the central product turns out to be the direct product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021190/c02119010.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021190/c02119011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021190/c02119012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021190/c02119013.png" /> are arbitrary groups such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021190/c02119014.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021190/c02119015.png" /> is a monomorphism, then the central product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021190/c02119016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021190/c02119017.png" /> can be defined without assuming in advance that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021190/c02119018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021190/c02119019.png" /> are subgroups of a certain group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021190/c02119020.png" />.
+
A group-theoretical construction. A group $G$ is called a central product of two of its subgroups $A$ and $B$ if it is generated by them, if $ab=ba$ for any two elements $a \in A$ and $b \in B$ and if the intersection $A \cap B$ lies in the [[Centre of a group|centre]] $\mathcal{Z}(G)$. In particular, for $A \cap B = \{1\}$ the central product turns out to be the [[direct product]] $A \times B$. If $A$, $B$ and $C$ are arbitrary groups such that $C \le \mathcal{Z}(A)$ and if $\theta : C \rightarrow \mathcal{Z}(B)$ is a monomorphism, then the (external) central product of $A$ and $B$ can be defined without assuming in advance that $A$ and $B$ are subgroups of a certain group $C$.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  D. Gorenstein,  "Finite groups" , Harper &amp; Row  (1968)</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  D. Gorenstein,  "Finite groups" , Harper &amp; Row  (1968)</TD></TR>
 +
</table>
 +
 
 +
{{TEX|done}}

Latest revision as of 21:01, 10 January 2017

A group-theoretical construction. A group $G$ is called a central product of two of its subgroups $A$ and $B$ if it is generated by them, if $ab=ba$ for any two elements $a \in A$ and $b \in B$ and if the intersection $A \cap B$ lies in the centre $\mathcal{Z}(G)$. In particular, for $A \cap B = \{1\}$ the central product turns out to be the direct product $A \times B$. If $A$, $B$ and $C$ are arbitrary groups such that $C \le \mathcal{Z}(A)$ and if $\theta : C \rightarrow \mathcal{Z}(B)$ is a monomorphism, then the (external) central product of $A$ and $B$ can be defined without assuming in advance that $A$ and $B$ are subgroups of a certain group $C$.

References

[1] D. Gorenstein, "Finite groups" , Harper & Row (1968)
How to Cite This Entry:
Central product of groups. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Central_product_of_groups&oldid=15113
This article was adapted from an original article by N.N. Vil'yams (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article