Join of Abelian and central implies Abelian

This article describes a computation relating the result of the Join operator (?) on two known subgroup properties (i.e., Abelian subgroup (?) and Central subgroup (?)), to another known subgroup property (i.e., Abelian subgroup (?))
View a complete list of join computations

Statement

Property-theoretic statement

$\langle$ Abelian, central $\rangle \leq$ 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.

Related facts

Converse

• Join-transiter of Abelian is central