Infimum of Subgroups in Lattice

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

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

Let $\struct {\mathbb G, \subseteq}$ be the complete lattice formed by $\mathbb G$ and $\subseteq$.

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

Then:
 * $\inf \set {H, K} = H \cap K$

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

From Set of Subgroups forms Complete Lattice:


 * $\inf \set {H, K} = H \cap K$