On normalizers of maximal subfields of division algebras

‎Here‎, ‎we investigate a conjecture posed by Amiri and Ariannejad claiming‎ ‎that if every maximal subfield of a division ring D has trivial normalizer‎, ‎then D is commutative‎. ‎Using Amitsur classification of‎ ‎finite subgroups of division rings‎, ‎it is essentially shown that if‎ ‎D is finite dimensional over its center then it contains a maximal‎ ‎subfield with non-trivial normalizer if and only if D∗ contains a‎ ‎non-abelian soluble subgroup‎. ‎This result generalizes a theorem of‎ ‎Mahdavi-Hezavehi and Tignol about cyclicity of division algebras of prime index.