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:
 * $$A \vee B \ \stackrel {\mathbf {def}} {=\!=} \ \left \langle {A \cup B}\right \rangle$$

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:


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

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:
 * $$\bigvee_{k=1}^n H_k \ \stackrel {\mathbf {def}} {=\!=} \ \left \langle {\bigcup_{k=1}^n H_k}\right \rangle$$

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

Also see

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