Union of Subgroups/Corollary 2

Theorem
Let $\left({G, \circ}\right)$ be a group, and let $H, K \le G$ such that neither $H \subseteq K$ nor $K \subseteq H$. Let $H \vee K$ be the join of $H$ and $K$.

Then $H \vee K = H \cup K$ iff $H \subseteq K$ or $K \subseteq H$.

Proof
From the definition of join, $H \vee K$ is the smallest subgroup of $G$ containing $H \cup K$.

The result follows from the main result.