Maximal implies modular

Verbal statement
Any maximal subgroup of a group is a modular subgroup.

Statement with symbols
Suppose $$A$$ is a maximal subgroup of $$G$$. Then, if $$B,C$$ are subgroups of $$G$$ such that $$A \le C$$, we have:

$$\langle A, B \cap C \rangle = \langle A,B \rangle \cap C$$.

Related facts

 * Permutable implies modular

Proof
Given: A group $$G$$, a maximal subgroup $$A$$.

To prove: If $$B,C$$ are subgroups of $$G$$ such that $$A \le C$$, then:

$$\langle A, B \cap C \rangle = \langle A,B \rangle \cap C$$.

Proof: Since $$A$$ is maximal, either $$C = A$$ or $$C = G$$. We consider both cases.

If $$A = C$$, then $$B \cap C \le A$$, so the left side is $$A$$. $$\langle A, B \rangle$$ contains $$A$$, so the right side is $$A$$. Thus, the identity holds for $$A = C$$.

If $$C = G$$, then both sides are $$\langle A, B$$, and again the identity holds.