Definition:Join of Subgroups/General Definition

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