Maximal permutable implies normal

This article gives the proof of a maximality equivalence. In other words, there are two subgroup properties: a stronger one (Normal subgroup (?)) and a weaker one (Permutable subgroup (?)). However, any subgroup maximal among the proper subgroups with the weaker property also has the stronger property, and is thus also maximal among proper subgroups with the stronger property.
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Statement

Suppose $G$ is a group and $H$ is maximal among the proper Permutable subgroup (?)s 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. Permutability is strongly join-closed
2. 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. $HK=⟨H,K⟩$ 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.

References

Textbook references

• Subnormal subgroups of groups by John C. Lennox and Stewart E. Stonehewer, Oxford Mathematical Monographs, ISBN 019853552X, Page 213, Theorem 7.1.1, ^{More info}