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.