Maximal permutable implies normal

Statement
Suppose $$G$$ is a group and $$H$$ is maximal among the proper fact about::permutable subgroups of $$G$$. Then, $$H$$ is a normal subgroup of $$G$$.

Related facts

 * Maximal conjugate-permutable implies normal
 * Pronormal implies self-conjugate-permutable
 * Maximal implies self-conjugate-permutable
 * Conjugate-permutable and self-conjugate-permutable implies normal

Applications

 * Permutable implies ascendant
 * Permutable implies subnormal in finite

Facts used

 * 1) uses::Permutability is strongly join-closed
 * 2) uses::Product of conjugates is proper

Proof
Given: A group $$G$$, a subgroup $$H$$ that is maximal among the proper permutable subgroups of $$G$$.

To prove: $$H$$ is normal in $$G$$: Any conjugate $$K$$ of $$H$$ in $$G$$ is contained in $$H$$.

Proof:
 * 1) $$K$$ is also permutable: This is because conjugation is an automorphism, so conjugate subgroups share the same properties.
 * 2) $$HK = \langle H, K \rangle$$ is also permutable: This follows from fact (1).
 * 3) $$HK \ne G$$: This follows from fact (2).
 * 4) $$HK = H$$: Since $$H \le HK \le G$$ and $$HK$$ is permutable, maximality of $$H$$ forces $$HK = H$$ or $$HK = G$$. The latter case is ruled out by the previous step, so $$HK = H$$.
 * 5) $$K \le H$$: This follows from the previous step.