Central product of groups

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)