Infimum of Subgroups in Lattice
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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:
- $\inf \set {H, K} = H \cap K$
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings: Theorem $14.6$