On normalizers of maximal subfields of division algebras

ýHereý, ýwe investigate a conjecture posed by Amiri and Ariannejad claimingý ýthat if every maximal subfield of a division ring D has trivial normalizerý, ýthen D is commutativeý. ýUsing Amitsur classification ofý ýfinite subgroups of division ringsý, ýit is essentially shown that ifý ýD is finite dimensional over its center then it contains a maximalý ýsubfield with non-trivial normalizer if and only if D∗ contains aý ýnon-abelian soluble subgroupý. ýThis result generalizes a theorem ofý ýMahdavi-Hezavehi and Tignol about cyclicity of division algebras of prime index.