Join of Subgroups is Group Generated by Union

Theorem
Let $G$ be a group.

Let $H$ and $K$ be subgroups of $G$.

Let $S$ be the set of words of $H \cup K$.

Then $S$ is a subgroup of $K$ such that:
 * $S = \gen {H \cup K} = H \vee K$

where $H \vee K$ denotes the join of $H$ and $K$.

Proof
By definition, the set of words in $H \cup K$ is:
 * $S = \map W {H \cup K} := \set {s_1 \circ s_2 \circ \cdots \circ s_n: n \in \N_{>0}: s_i \in H \cup K 1 \le i \le n}$

Let $h \in H$.

Then setting $n = 1$ in the above definition and letting $s_1 = h$ it follows that $H \subseteq S$.

Similarly it is seen that $K \subseteq S$

So from Union is Smallest Superset it follows that $H \cup K \subseteq S$.

From Set of Words Generates Group it follows that $S$ is the subgroup of $G$ generated by $H \cup K$.