\begin{definition}[Definition:Join of Subgroups/General Definition]
Let $\struct {G, \circ}$ be a group.
Let $H_1, H_2, \ldots, H_n$ be subgroups of $G$.
Then the '''join''' of $H_1, H_2, \ldots, H_n$ is defined as:
:$\ds \bigvee_{k \mathop = 1}^n H_k := \gen {\bigcup_{k \mathop = 1}^n H_k}$
or:
:$\ds \bigvee_{k \mathop = 1}^n H_k := \bigcap \set {T: T \text { is a subgroup of } G: \bigcup_{k \mathop = 1}^n H_k \subseteq T}$
\end{definition}