Permutable implies modular

Statement
Any permutable subgroup of a group is modular.

Permutable subgroup
A subgroup $$H$$ of a group $$G$$ is termed permutable if $$HK = KH = \langle H, K \rangle$$ for every subgroup $$K \le G$$.

Facts used

 * 1) Modular property of groups: This states that if $$H,K,L$$ are subgroups of $$G$$ such that $$H \le L$$, then:

$$H(K \cap L) = HK \cap L$$.

Proof
Given: A subgroup $$H$$ of a group $$G$$ such that $$HK = KH = \langle H, K \rangle$$ for all subgroups $$K \le G$$.

To prove: For any subgroups $$K, L$$ of $$G$$ such that $$H \le L$$, we have:

$$\langle H, K \cap L \rangle = \langle H, K \rangle \cap L$$.

Proof: Since $$H$$ is permutable, we have:

$$\langle H, K \cap L \rangle = H(K \cap L)$$.

and:

$$\langle H, K \rangle = HK$$, so $$\langle H, K \rangle \cap L = HK \cap L$$.

Applying fact (1) now yields the result.