Definition:Join of Subgroups

Definition
Let $\left({G, \cdot}\right)$ be a group.

Let $A$ and $B$ be subgroups of $G$.

The join of $A$ and $B$ is written and defined as:
 * $A \vee B := \left \langle {A \cup B}\right \rangle$

where $\left \langle {A \cup B}\right \rangle$ is the group generated by $A \cup B$.

By the definition of group generator, this can alternatively be written:


 * $\displaystyle A \vee B := \bigcap \left\{{T: T \text { is a subgroup of } G: A \cup B \subseteq T}\right\}$

General Definition
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:
 * $\displaystyle \bigvee_{k=1}^n H_k := \left \langle {\bigcup_{k=1}^n H_k}\right \rangle$

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

Also see

 * Union of Subgroups, where it is shown that $A \vee B = A \cup B$ iff $A \subseteq B$ or $B \subseteq A$.