# Definition:Join of Subgroups

## Definition

Let $\struct {G, \circ}$ be a group.

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

The join of $A$ and $B$ is written and defined as:

$A \vee B := \gen {A \cup B}$

where $\gen {A \cup B}$ is the subgroup generated by $A \cup B$.

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

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

### 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:

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

or:

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

## Also see

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