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