Infimum and Supremum of Subgroups

Theorem
Let $\left({G, \circ}\right)$ be a group.

Let $\mathbb G$ be the set of all subgroups of $G$.

Let $\left({\mathbb G, \subseteq}\right)$ be the complete lattice formed by $\mathbb G$ and $\subseteq$.

Let $H, K \in \mathbb G$.

Then:
 * $\inf \left\{{H, K}\right\} = H \cap K$;
 * If either $H$ or $K$ are normal in $G$, then:
 * $\sup \left\{{H, K}\right\} = H \circ K$
 * where $H \circ K$ denotes subset product.

Proof
Let $H, K \in \mathbb G$.


 * From complete lattice we have:
 * $\inf \left\{{H, K}\right\} = H \cap K$


 * Now let $L = \sup \left\{{H, K}\right\}$, and let either $H$ or $K$ be normal in $G$.

The smallest subgroup of $G$ containing $H$ and $K$ is $\left \langle {H, K} \right \rangle$, the subgroup generated by $H$ and $K$.

From Subgroup Product with Normal Subgroup as Generator, $\left \langle {H, K} \right \rangle = H \circ K$ when either $H$ or $K$ is normal.

The result follows.