\begin{definition}[Definition:Join of Subgroups]
Let $\struct {G, \circ}$ be a group.
Let $A$ and $B$ be subgroups of $G$.
The '''join''' of $A$ and $B$ is written and defined as:
:$A \vee B := \gen {A \cup B}$
where $\gen {A \cup B}$ is the subgroup generated by $A \cup B$.
By the definition of subgroup generator, this can alternatively be written:
:$\ds A \vee B := \bigcap \set {T: T \text { is a subgroup of } G: A \cup B \subseteq T}$
\end{definition}