Infimum and Supremum of Subgroups

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 $H, K \in \mathbb G$.

Infimum of Subgroups in Lattice

Then:

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

Supremum of Subgroups in Lattice

Let either $H$ or $K$ be normal in $G$.

Then:

$\sup \set {H, K} = H \circ K$

where $H \circ K$ denotes subset product.

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings: Theorem $14.6$