Definition:Join of Subgroups/General Definition

Definition
Let $\left({G, \circ}\right)$ 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:
 * $\displaystyle \bigvee_{k \mathop = 1}^n H_k := \left \langle {\bigcup_{k \mathop = 1}^n H_k}\right \rangle$

or:
 * $\displaystyle \bigvee_{k \mathop = 1}^n H_k := \bigcap \left\{{T: T \text { is a subgroup of } G: \bigcup_{k \mathop = 1}^n H_k \subseteq T}\right\}$