Join of Abelian and central implies Abelian

Property-theoretic statement
$$\langle$$ Abelian, central $$\rangle \le$$ Abelian

Statement with symbols
Suppose $$G$$ is a group, $$H,K$$ are subgroups, with $$H$$ an Abelian subgroup (i.e., $$H$$ is Abelian as a group) and $$K$$ a central subgroup (i.e., $$K$$ is contained in the center of $$G$$). Then, the join of subgroups $$\langle H,K \rangle$$ (i.e., the subgroup generated by $$H$$ and $$K$$) is Abelian.

Converse

 * Join-transiter of Abelian is central