Join of direct factor and central subgroup

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed a join of direct factor and central subgroup or a central subgroup over direct factor if it satisfies the following equivalent conditions:


 * 1) There exist subgroups $$A,B$$ of $$G$$ such that $$A$$ is a defining ingredient::direct factor of $$G$$, $$B$$ is a defining ingredient::central subgroup of $$G$$, and $$H$$ is the join (in this case, also the product) of $$A$$ and $$B$$.
 * 2) There exists a direct factor $$A$$ of $$G$$ contained in $$H$$ such that $$H/A$$ is a central subgroup of the quotient group $$G/A$$.