Difference between revisions of "Central product of groups"
From Encyclopedia of Mathematics
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| − | A group-theoretical construction. A group | + | 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 & Row (1968)</TD></TR></table> | + | <table> |
| + | <TR><TD valign="top">[1]</TD> <TD valign="top"> D. Gorenstein, "Finite groups" , Harper & Row (1968)</TD></TR> | ||
| + | </table> | ||
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| + | {{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
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