Intersection of Subgroups is Subgroup/General Result

Theorem
Let $\struct {G, \circ}$ be a group. Let $\mathbb S$ be a set of subgroups of $\struct {G, \circ}$, where $\mathbb S \ne \O$.

Then the intersection $\ds \bigcap \mathbb S$ of the elements of $\mathbb S$ is itself a subgroup of $G$.

Also, $\ds \bigcap \mathbb S$ is the largest subgroup of $\struct {G, \circ}$ contained in each element of $\mathbb S$.

Proof
Let $\ds H = \bigcap \mathbb S$.

Let $H_k$ be an arbitrary element of $\mathbb S$.

Then:

Now to show that $\struct {H, \circ}$ is the largest such subgroup.

Let $K$ be a subgroup of $\struct {G, \circ}$ such that:
 * $\forall S \in \mathbb S: K \subseteq S$

Then by definition $K \subseteq H$.

Let $x, y \in K$.

Then:
 * $x \circ y^{-1} \in K \implies x \circ y^{-1} \in H$

Thus any subgroup of all elements of $\mathbb S$ is also a subgroup of $H$ and so no larger than $H$.

Thus $\ds H = \bigcap \mathbb S$ is the largest subgroup of $S$ contained in each element of $\mathbb S$.